When you start GAP it already knows several groups. Currently GAP initially knows the following groups: item some basic groups, such as cyclic groups or symmetric groups (see The Basic Groups Library), The Primitive Groups Library), The Transitive Groups Library), The Solvable Groups Library), item the 2-groups of size at most 256 (see The 2-Groups Library), item the 3-groups of size at most 729 (see The 3-Groups Library), item the irreducible solvable subgroups of GL(n,p) for n > 1 and The Irreducible Solvable Linear Groups Library), item the finite perfect groups of size at most 10^6 (excluding 11 sizes) (see The Library of Finite Perfect Groups), item the irreducible maximal finite integral matrix groups of dimension Irreducible Maximal Finite Integral Matrix Groups), The Crystallographic Groups Library).
Each of the set of groups above is called a library. The whole set of groups that GAP knows initially is called the GAP collection of group libraries. There is usually no relation between the groups in the different libraries.
Several of the libraries are accessed in a uniform manner. For each of these libraries there is a so called selection function that allows you to select the list of groups that satisfy given criterias from a library. The example function allows you to select one group that satisfies given criteria from the library. The low-level extraction function allows you to extract a single group from a library, using a simple Selection Functions, Example Functions, and Extraction Functions.
Note that a system administrator may choose to install all, or only a few, or even none of the libraries. So some of the libraries mentioned below may not be available on your installation.
CyclicGroup( n )
CyclicGroup( D, n )
In the first form CyclicGroup
returns the cyclic group of size n as a
permutation group. In the second form D must be a domain of group
elements, e.g., Permutations
or AgWords
, and CyclicGroup
returns
the cyclic group of size n as a group of elements of that type.
gap> c12 := CyclicGroup( 12 ); Group( ( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12) ) gap> c105 := CyclicGroup( AgWords, 5*3*7 ); Group( c105_1, c105_2, c105_3 ) gap> Order(c105,c105.1); Order(c105,c105.2); Order(c105,c105.3); 105 35 7
AbelianGroup( sizes )
AbelianGroup( D, sizes )
In the first form AbelianGroup
returns the abelian group C_{sizes[1]}
* C_{sizes[2]} * ... * C_{sizes[n]}, where sizes must be a list
of positive integers, as a permutation group. In the second form D
must be a domain of group elements, e.g., Permutations
or AgWords
,
and AbelianGroup
returns the abelian group as a group of elements of
this type.
gap> g := AbelianGroup( AgWords, [ 2, 3, 7 ] ); Group( a, b, c ) gap> Size( g ); 42 gap> IsAbelian( g ); true
The default function GroupElementsOps.AbelianGroup
uses the functions
CyclicGroup
and DirectProduct
(see DirectProduct) to construct the
abelian group.
ElementaryAbelianGroup( n )
ElementaryAbelianGroup( D, n )
In the first form ElementaryAbelianGroup
returns the elementary abelian
group of size n as a permutation group. n must be a positive prime
power of course. In the second form D must be a domain of group
elements, e.g., Permutations
or AgWords
, and ElementaryAbelianGroup
returns the elementary abelian group as a group of elements of this type.
gap> ElementaryAbelianGroup( 16 ); Group( (1,2), (3,4), (5,6), (7,8) ) gap> ElementaryAbelianGroup( AgWords, 3 ^ 10 ); Group( m59049_1, m59049_2, m59049_3, m59049_4, m59049_5, m59049_6, m59049_7, m59049_8, m59049_9, m59049_10 )
The default function GroupElementsOps.ElementaryAbelianGroup
uses
CyclicGroup
and DirectProduct
(see DirectProduct to construct the
elementary abelian group.
DihedralGroup( n )
DihedralGroup( D, n )
In the first form DihedralGroup
returns the dihedral group of size n
as a permutation group. n must be a positive even integer. In the
second form D must be a domain of group elements, e.g., Permutations
or AgWords
, and DihedralGroup
returns the dihedral group as a group
of elements of this type.
gap> DihedralGroup( 12 ); Group( (1,2,3,4,5,6), (2,6)(3,5) )
PolyhedralGroup( p, q )
PolyhedralGroup( D, p, q )
In the first form PolyhedralGroup
returns the polyhedral group of size
p * q
as a permutation group. p and q must be positive
integers and there must exist a nontrivial p-th root of unity modulo
every prime factor of q. In the second form D must be a domain of
group elements, e.g., Permutations
or Words
, and PolyhedralGroup
returns the polyhedral group as a group of elements of this type.
gap> PolyhedralGroup( 3, 13 ); Group( ( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13), ( 2, 4,10)( 3, 7, 6) ( 5,13,11)( 8, 9,12) ) gap> Size( last ); 39
SymmetricGroup( d )
SymmetricGroup( D, d )
In the first form SymmetricGroup
returns the symmetric group of degree
d as a permutation group. d must be a positive integer. In the
second form D must be a domain of group elements, e.g., Permutations
or Words
, and SymmetricGroup
returns the symmetric group as a group
of elements of this type.
gap> SymmetricGroup( 8 ); Group( (1,8), (2,8), (3,8), (4,8), (5,8), (6,8), (7,8) ) gap> Size( last ); 40320
AlternatingGroup( d )
AlternatingGroup( D, d )
In the first form AlternatingGroup
returns the alternating group of
degree d as a permutation group. d must be a positive integer. In
the second form D must be a domain of group elements, e.g.,
Permutations
or Words
, and AlternatingGroup
returns the alternating
group as a group of elements of this type.
gap> AlternatingGroup( 8 ); Group( (1,2,8), (2,3,8), (3,4,8), (4,5,8), (5,6,8), (6,7,8) ) gap> Size( last ); 20160
GeneralLinearGroup( n, q )
GeneralLinearGroup( D, n, q )
In the first form GeneralLinearGroup
returns the general linear group
GL( <n>, <q> ) as a matrix group. In the second form D must be a
domain of group elements, e.g., Permutations
or AgWords
, and
GeneralLinearGroup
returns GL( <n>, <q> ) as a group of elements of
that type.
gap> g := GeneralLinearGroup( 2, 4 ); Size( g ); GL(2,4) 180
SpecialLinearGroup( n, q )
SpecialLinearGroup( D, n, q )
In the first form SpecialLinearGroup
returns the special linear group
SL( <n>, <q> ) as a matrix group. In the second form D must be a
domain of group elements, e.g., Permutations
or AgWords
, and
SpecialLinearGroup
returns SL( <n>, <q> ) as a group of elements of
that type.
gap> g := SpecialLinearGroup( 3, 4 ); Size( g ); SL(3,4) 60480
SymplecticGroup( n, q )
SymplecticGroup( D, n, q )
In the first form SymplecticGroup
returns the symplectic group SP(
<n>, <q> ) as a matrix group. In the second form D must be a domain
of group elements, e.g., Permutations
or AgWords
, and
SymplecticGroup
returns SP( <n>, <q> ) as a group of elements of that
type.
gap> g := SymplecticGroup( 4, 2 ); Size( g ); SP(4,2) 720
GeneralUnitaryGroup( n, q )
GeneralUnitaryGroup( D, n, q )
In the first form GeneralUnitaryGroup
returns the general unitary group
GU( <n>, <q> ) as a matrix group. In the second form D must be a
domain of group elements, e.g., Permutations
or AgWords
, and
GeneralUnitaryGroup
returns GU( <n>, <q> ) as a group of elements of
that type.
gap> g := GeneralUnitaryGroup( 3, 3 ); Size( g ); GU(3,3) 24192
SpecialUnitaryGroup( n, q )
SpecialUnitaryGroup( D, n, q )
In the first form SpecialUnitaryGroup
returns the special unitary group
SU( <n>, <q> ) as a matrix group. In the second form D must be a
domain of group elements, e.g., Permutations
or AgWords
, and
SpecialUnitaryGroup
returns SU( <n>, <q> ) as a group of elements of
that type.
gap> g := SpecialUnitaryGroup( 3, 3 ); Size( g ); SU(3,3) 6048
MathieuGroup( d )
MathieuGroup
returns the Mathieu group of degree d as a permutation
group. d is expected to be 11, 12, 22, 23, or 24.
gap> g := MathieuGroup( 12 ); Size( g ); Group( ( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11), ( 3, 7,11, 8) ( 4,10, 5, 6), ( 1,12)( 2,11)( 3, 6)( 4, 8)( 5, 9)( 7,10) ) 95040
AllLibraryGroups( fun1, val1, fun2, val2, ... )
For each group library there is a selection function. This function allows you to select all groups from the library that have a given set of properties.
The name of the selection functions always begins with All
and always
ends with Groups
. Inbetween is a name that hints at the nature of the
group library. For example, the selection function for the library of all
The Primitive Groups Library) is called AllPrimitiveGroups
, and the selection function for
The 2-Groups Library) is called AllTwoGroups
.
These functions take an arbitrary number of pairs of arguments. The first argument in such a pair is a function that can be applied to the groups in the library, and the second argument is either a single value that this function must return in order to have this group included in the selection, or a list of such values.
For example
AllPrimitiveGroups( DegreeOperation, [10..15], Size, [1..100], IsAbelian, false );
should return a list of all primitive groups with degree between 10 and 15 and size less than 100 that are not abelian.
Thus the AllPrimitiveGroups
behaves as if it was implemented by a
function similar to the one defined below, where PrimitiveGroupsList
is
a list of all primitive groups. Note, in the definition below we assume
for simplicity that AllPrimitiveGroups
accepts exactly 4 arguments. It
is of course obvious how to change this definition so that the function
would accept a variable number of arguments.
AllPrimitiveGroups := function ( fun1, val1, fun2, val2 ) local groups, g, i; groups := []; for i in [ 1 .. Length( PrimitiveGroupsList ) ] do g := PrimitiveGroupsList[i]; if fun1(g) = val1 or IsList(val1) and fun1(g) in val1 and fun2(g) = val2 or IsList(val2) and fun2(g) in val2 then Add( groups, g ); fi; od; return groups; end;
Note that the real selection functions are considerably more difficult, to improve the efficiency. Most important, each recognizes a certain set About Group Libraries).
OneLibraryGroup( fun1, val1, fun2, val2, ... )
For each group library there is a example function. This function allows you to find one group from the library that has a given set of properties.
The name of the example functions always begins with One
and always
ends with Group
. Inbetween is a name that hints at the nature of the
group library. For example, the example function for the library of all
The Primitive Groups Library) is called OnePrimitiveGroup
, and the example function for the
library of all 2-groups of size at most 256 (see The 2-Groups Library)
is called OneTwoGroup
.
These functions take an arbitrary number of pairs of arguments. The first argument in such a pair is a function that can be applied to the groups in the library, and the second argument is either a single value that this function must return in order to have this group returned by the example function, or a list of such values.
For example
OnePrimitiveGroup( DegreeOperation, [10..15], Size, [1..100], IsAbelian, false );
should return one primitive group with degree between 10 and 15 and size size less than 100 that is not abelian.
Thus the OnePrimitiveGroup
behaves as if it was implemented by a
function similar to the one defined below, where PrimitiveGroupsList
is
a list of all primitive groups. Note, in the definition below we assume
for simplicity that OnePrimitiveGroup
accepts exactly 4 arguments. It
is of course obvious how to change this definition so that the function
would accept a variable number of arguments.
OnePrimitiveGroup := function ( fun1, val1, fun2, val2 ) local g, i; for i in [ 1 .. Length( PrimitiveGroupsList ) ] do g := PrimitiveGroupsList[i]; if fun1(g) = val1 or IsList(val1) and fun1(g) in val1 and fun2(g) = val2 or IsList(val2) and fun2(g) in val2 then return g; fi; od; return false; end;
Note that the real example functions are considerably more difficult, to improve the efficiency. Most important, each recognizes a certain set of About Group Libraries).
For each group library there is an extraction function. This function allows you to extract single groups from the library.
The name of the extraction function always ends with Group
and begins
with a name that hints at the nature of the library. For example the
The Primitive Groups Library) is called PrimitiveGroup
, and the extraction
The 2-Groups Library) is called TwoGroup
.
What arguments the extraction function accepts, and how they are interpreted is described in the sections that describe the individual group libraries.
For example
PrimitiveGroup( 10, 4 );
returns the 4-th primitive group of degree 10.
The reason for the extraction function is as follows. A group library is usually not stored as a list of groups. Instead a more compact representation for the groups is used. For example the groups in the library of 2-groups are represented by 4 integers. The extraction function hides this representation from you, and allows you to access the group library as if it was a table of groups (two dimensional in the above example).
This group library contains all primitive permutation groups of degree at most 50. There are a total of 406 such groups. Actually to be a little bit more precise, there are 406 inequivalent primitive operations on at most 50 points. Quite a few of the 406 groups are isomorphic.
AllPrimitiveGroups( fun1, val1, fun2, val2, ... )
AllPrimitiveGroups
returns a list containing all primitive groups that
have the properties given as arguments. Each property is specified by
passing a pair of arguments, the first being a function, which will be
applied to all groups in the library, and the second being a value or a
list of values, that this function must return in order to have this
group included in the list returned by AllPrimitiveGroups
.
The first argument must be DegreeOperation
and the second argument
either a degree or a list of degrees, otherwise AllPrimitiveGroups
will
print a warning to the effect that the library contains only groups with
degrees between 1 and 50.
gap> l := AllPrimitiveGroups( Size, 120, IsSimple, false ); #W AllPrimitiveGroups: degree automatically restricted to [1..50] [ S(5), PGL(2,5), S(5) ] gap> List( l, g -> g.generators ); [ [ (1,2,3,4,5), (1,2) ], [ (1,2,3,4,5), (2,3,5,4), (1,6)(3,4) ], [ ( 1, 8)( 2, 5, 6, 3)( 4, 9, 7,10), ( 1, 5, 7)( 2, 9, 4)( 3, 8,10) ] ]
OnePrimitiveGroup( fun1, val1, fun2, val2, ... )
OnePrimitiveGroup
returns one primitive group that has the properties
given as argument. Each property is specified by passing a pair of
arguments, the first being a function, which will be applied to all
groups in the library, and the second being a value or a list of values,
that this function must return in order to have this group returned by
OnePrimitiveGroup
. If no such group exists, false
is returned.
The first argument must be DegreeOperation
and the second argument
either a degree or a list of degrees, otherwise OnePrimitiveGroup
will
print a warning to the effect that the library contains only groups with
degrees between 1 and 50.
gap> g := OnePrimitiveGroup( DegreeOperation,5, IsSolvable,false ); A(5) gap> Size( g ); 60
AllPrimitiveGroups
and OnePrimitiveGroup
recognize the following
functions and handle them usually quite efficient. DegreeOperation
,
Size
, Transitivity
, and IsSimple
. You should pass those functions
first, e.g., it is more efficient to say AllPrimitiveGroups( Size,120 ,
IsAbelian,false )
than to say AllPrimitiveGroups( IsAbelian,false,
Size,120 )
(see About Group Libraries).
PrimitiveGroup( deg, nr )
PrimitiveGroup
returns the nr-th primitive group of degree deg.
Both deg and nr must be positive integers. The primitive groups of
equal degree are sorted with respect to their size, so for example
PrimitiveGroup( deg, 1 )
is the smallest primitive group of degree
deg, e.g, the cyclic group of size deg, if deg is a prime.
Primitive groups of equal degree and size are in no particular order.
gap> g := PrimitiveGroup( 8, 1 ); AGL(1,8) gap> g.generators; [ (1,2,3,4,5,6,7), (1,8)(2,4)(3,7)(5,6) ]
Apart from the usual components described in Group Records, the group records returned by the above functions have the following components.
transitivity
:
isSharpTransitive
:true
if G is sharply G.transitivity
-fold transitive and
false
otherwise.
isKPrimitive
:true
if G is k-fold primitive, and false
otherwise.
isOdd
:false
if G is a subgroup of the alternating group of degree
G.degree
and true
otherwise.
isFrobeniusGroup
:true
if G is a Frobenius group indexFrobenius group and
false
otherwise.
This library was computed by Charles Sims. The list of primitive permutation groups of degree at most 20 was published in Sim70. The library was brought into GAP format by Martin Schaccent127onert. He assumes the responsibility for all mistakes.
The transitive groups library contains representatives for all transitive permutation groups of degree at most 22. Two permutations groups of the same degree are considered to be equivalent, if there is a renumbering of points, which maps one group into the other one. In other words, if they lie in the save conjugacy class under operation of the full symmetric group by conjugation.
There are a total of 4945 such groups up to degree 22.
AllTransitiveGroups( fun1, val1, fun2, val2, ... )
AllTransitiveGroups
returns a list containing all transitive groups
that have the properties given as arguments. Each property is specified
by passing a pair of arguments, the first being a function, and the
second being a value or a list of values. AllTransitiveGroups
will
return all groups from the transitive groups library, for which all
specified functions have the specified values.
If the degree is not restricted to 22 at most, AllTransitiveGroups
will
print a warning.
OneTransitiveGroup( fun1, val1, fun2, val2, ... )
OneTransitiveGroup
returns one transitive group that has the properties
given as argument. Each property is specified by passing a pair of
arguments, the first being a function, and the second being a value or a
list of values. OneTransitiveGroup
will return one groups from the
transitive groups library, for which all specified functions have the
specified values. If no such group exists, false
is returned.
If the degree is not restricted to 22 at most, OneTransitiveGroup
will
print a warning.
AllTransitiveGroups
and OneTransitiveGroup
recognize the following
functions and get the corresponding properties from a precomputed list to
speed up processing:
DegreeOperation
, Size
, Transitivity
, and
IsPrimitive
. You do not need to pass those functions first, as the
selection function picks the these properties first.
TransitiveGroup( deg, nr )
TransitiveGroup
returns the nr-th transitive group of degree deg.
Both deg and nr must be positive integers. The transitive groups of
equal degree are sorted with respect to their size, so for example
TransitiveGroup( deg, 1 )
is the smallest transitive group of degree
deg, e.g, the cyclic group of size deg, if deg is a prime. The
ordering of the groups corresponds to the list in Butler/McKay
BM83.
This library was computed by Gregory Butler, John McKay, Gordon Royle and Alexander Hulpke. The list of transitive groups up to degree 11 was published in BM83, the list of degree 12 was published in Roy87, degree 14 and 15 were published in But93.
The library was brought into GAP format by Alexander Hulpke, who is responsible for all mistakes.
gap> TransitiveGroup(10,22); S(5)[x]2 gap> l:=AllTransitiveGroups(DegreeOperation,12,Size,1440, > IsSolvable,false); [ S(6)[x]2, M_10.2(12) = A_6.E_4(12) = [S_6[1/720]{M_10}S_6]2 ] gap> List(l,IsSolvable); [ false, false ]
TransitiveIdentification( G )
Let G be a permutation group, acting transitively on a set of up to 22
points. Then TransitiveIdentification
will return the position of this
group in the transitive groups library. This means, if G operates on
m points and TransitiveIdentification
returns n, then G is
permutation isomorphic to the group TransitiveGroup(m,n)
.
gap> TransitiveIdentification(Group((1,2),(1,2,3))); 2
GAP has a library of the 1045 solvable groups of size between 2 and 100. The groups are from lists computed by M.~Hall and J.~K.~Senior (size 64, see HS64), R.~Laue (size 96, see Lau82) and J.~Neubaccent127user (other sizes, see Neu67).
AllSolvableGroups( fun1, val1, fun2, val2, ... )
AllSolvableGroups
returns a list containing all solvable groups that
have the properties given as arguments. Each property is specified by
passing a pair of arguments, the first being a function, which will be
applied to all the groups in the library, and the second being a value or
a list of values, that this function must return in order to have this
group included in the list returned by AllSolvableGroups
.
gap> AllSolvableGroups(Size,24,IsNontrivialDirectProduct,false); [ 12.2, grp_24_11, D24, Q8+S3, Sl(2,3), S4 ]
OneSolvableGroup( fun1, val1, fun2, val2, ... )
OneSolvableGroup
returns a solvable group with the specified
properties. Each property is specified by passing a pair of arguments,
the first being a function, which will be applied to all the groups in
the library, and the second being a value or a list of values, that this
function must return in order to have this group returned by
OneSolvableGroup
. If no such group exists, false
is returned.
gap> OneSolvableGroup(Size,100,x->Size(DerivedSubgroup(x)),10); false gap> OneSolvableGroup(Size,24,IsNilpotent,false); S3x2^2
AllSolvableGroups
and OneSolvableGroup
recognize the following
functions and handle them usually very efficiently: Size
, IsAbelian
,
IsNilpotent
, and IsNonTrivialDirectProduct
.
SolvableGroup( size, nr )
SolvableGroup
returns the nr-th group of size size in the library.
SolvableGroup
will signal an error if size is not between 2 and 100,
or if nr is larger than the number of solvable groups of size size.
Finite Polycyclic Groups).
gap> SolvableGroup( 32 , 15 ); Q8x4
The library of 2-groups contains all the 2-groups of size dividing 256. There are a total of 58760 such groups, 1 of size 2, 2 of size 4, 5 of size 8, 14 of size 16, 51 of size 32, 267 of size 64, 2328 of size 128, and 56092 of size 256.
AllTwoGroups( fun1, val1, fun2, val2, ... )
AllTwoGroups
returns the list of all the 2-groups that have the
properties given as arguments. Each property is specified by passing a
pair of arguments, the first is a function that can be applied to each
group, the second is either a single value or a list of values that the
function must return in order to select that group.
gap> l := AllTwoGroups( Size, 256, Rank, 3, pClass, 2 ); [ Group( a1, a2, a3, a4, a5, a6, a7, a8 ), Group( a1, a2, a3, a4, a5, a6, a7, a8 ), Group( a1, a2, a3, a4, a5, a6, a7, a8 ), Group( a1, a2, a3, a4, a5, a6, a7, a8 ) ] gap> List( l, g -> Length( ConjugacyClasses( g ) ) ); [ 112, 88, 88, 88 ]
OneTwoGroup( fun1, val1, fun2, val2, ... )
OneTwoGroup
returns a single 2-group that has the properties given as
arguments. Each property is specified by passing a pair of arguments,
the first is a function that can be applied to each group, the second is
either a single value or a list of values that the function must return
in order to select that group.
gap> g := OneTwoGroup( Size, [64..128], Rank, [2..3], pClass, 5 ); #I size restricted to [ 64, 128 ] Group( a1, a2, a3, a4, a5, a6 ) gap> Size( g ); 64 gap> Rank( g ); 2
AllTwoGroups
and OneTwoGroup
recognize the following functions and
handle them usually very efficiently. Size
, Rank
for the rank of the
Frattini quotient of the group, and pClass
for the exponent-p class
of the group. Note that Rank
and pClass
are dummy functions that
can be used only in this context, i.e., they can not be applied to
arbitrary groups.
TwoGroup( size, nr )
TwoGroup
returns the nr-th group of size size. The group is
returned as a finite polycyclic group (see Finite Polycyclic Groups).
TwoGroup
will signal an error if size is not a power of 2 between 2
and 256, or nr is larger than the number of groups of size size.
Within each size the following criteria have been used, in turn, to determine the index position of a group in the list
gap> g := TwoGroup( 32, 45 ); Group( a1, a2, a3, a4, a5 ) gap> Rank( g ); 4 gap> pClass( g ); 2 gap> g.abstractRelators; [ a1^2*a5^-1, a2^2, a2^-1*a1^-1*a2*a1, a3^2, a3^-1*a1^-1*a3*a1, a3^-1*a2^-1*a3*a2, a4^2, a4^-1*a1^-1*a4*a1, a4^-1*a2^-1*a4*a2, a4^-1*a3^-1*a4*a3, a5^2, a5^-1*a1^-1*a5*a1, a5^-1*a2^-1*a5*a2, a5^-1*a3^-1*a5*a3, a5^-1*a4^-1*a5*a4 ]
Apart from the usual components described in Group Records, the group records returned by the above functions have the following components.
rank
:
pclass
:
abstractGenerators
:
abstractRelators
:Descriptions of the algorithms used in constructing the library data may be found in citeOBr90,OBr91. Using these techniques, a library was first prepared in 1987 by M.F.~Newman and E.A.~O'Brien; a partial description may be found in NO89.
The library was brought into the GAP format by Werner Nickel, Alice Niemeyer, and E.A.~O'Brien.
The library of 3-groups contains all the 3-groups of size dividing 729. There are a total of 594 such groups, 1 of size 3, 2 of size 9, 5 of size 27, 15 of size 81, 67 of size 243, and 504 of size 729.
AllThreeGroups( fun1, val1, fun2, val2, ... )
AllThreeGroups
returns the list of all the 3-groups that have the
properties given as arguments. Each property is specified by passing a
pair of arguments, the first is a function that can be applied to each
group, the second is either a single value or a list of values that the
function must return in order to select that group.
gap> l := AllThreeGroups( Size, 243, Rank, [2..4], pClass, 3 );; gap> Length ( l ); 33 gap> List( l, g -> Length( ConjugacyClasses( g ) ) ); [ 35, 35, 35, 35, 35, 35, 35, 243, 99, 99, 51, 51, 51, 51, 51, 51, 51, 51, 99, 35, 243, 99, 99, 51, 51, 51, 51, 51, 35, 35, 35, 35, 35 ]
OneThreeGroup( fun1, val1, fun2, val2, ... )
OneThreeGroup
returns a single 3-group that has the properties given as
arguments. Each property is specified by passing a pair of arguments,
the first is a function that can be applied to each group, the second is
either a single value or a list of values that the function must return
in order to select that group.
gap> g := OneThreeGroup( Size, 729, Rank, 4, pClass, [3..5] ); Group( a1, a2, a3, a4, a5, a6 ) gap> IsAbelian( g ); true
AllThreeGroups
and OneThreeGroup
recognize the following functions
and handle them usually very efficiently. Size
, Rank
for the rank of
the Frattini quotient of the group, and pClass
for the exponent-p
class of the group. Note that Rank
and pClass
are dummy functions
that can be used only in this context, i.e., they cannot be applied to
arbitrary groups.
ThreeGroup( size, nr )
ThreeGroup
returns the nr-th group of size size. The group is
returned as a finite polycyclic group (see Finite Polycyclic Groups).
ThreeGroup
will signal an error if size is not a power of 3 between
3 and 729, or nr is larger than the number of groups of size size.
Within each size the following criteria have been used, in turn, to determine the index position of a group in the list
gap> g := ThreeGroup( 243, 56 ); Group( a1, a2, a3, a4, a5 ) gap> pClass( g ); 3 gap> g.abstractRelators; [ a1^3, a2^3, a2^-1*a1^-1*a2*a1*a4^-1, a3^3, a3^-1*a1^-1*a3*a1, a3^-1*a2^-1*a3*a2*a5^-1, a4^3, a4^-1*a1^-1*a4*a1*a5^-1, a4^-1*a2^-1*a4*a2, a4^-1*a3^-1*a4*a3, a5^3, a5^-1*a1^-1*a5*a1, a5^-1*a2^-1*a5*a2, a5^-1*a3^-1*a5*a3, a5^-1*a4^-1*a5*a4 ]
Apart from the usual components described in Group Records, the group records returned by the above functions have the following components.
rank
:
pclass
:
abstractGenerators
:
abstractRelators
:Descriptions of the algorithms used in constructing the library data may be found in citeOBr90,OBr91.
The library was generated and brought into GAP format by E.A.~O'Brien and Colin Rhodes. David Baldwin, M.F.~Newman, and Maris Ozols have contributed in various ways to this project and to correctly determining these groups. The library design is modelled on and borrows extensively from the 2-groups library, which was brought into GAP format by Werner Nickel, Alice Niemeyer, and E.A.~O'Brien.
The IrredSol group library provides access to the irreducible solvable subgroups of GL(n,p), where n > 1, p is prime and p^n < 256. The library contains exactly one member from each of the 370 conjugacy classes of such subgroups.
By well known theory, this library also doubles as a library of primitive
solvable permutation groups of non-prime degree less than 256. To access
the data in this form, you must first build the matrix group(s) of
interest and then call the function
PrimitivePermGroupIrreducibleMatGroup( matgrp )
This function returns a permutation group isomorphic to the semidirect
product of an irreducible matrix group (over a finite field) and its
underlying vector space.
AllIrreducibleSolvableGroups( fun1, val1, fun2, val2, ... )
AllIrreducibleSolvableGroups
returns a list containing all irreducible
solvable linear groups that have the properties given as arguments. Each
property is specified by passing a pair of arguments, the first being a
function which will be applied to all groups in the library, and the
second being a value or a list of values that this function must return
in order to have this group included in the list returned by
AllIrreducibleSolvableGroups
.
gap> AllIrreducibleSolvableGroups( Dimension, 2, > CharFFE, 3, > Size, 8 ); [ Group( [ [ 0*Z(3), Z(3)^0 ], [ Z(3)^0, 0*Z(3) ] ], [ [ Z(3), 0*Z(3) ], [ 0*Z(3), Z(3)^0 ] ], [ [ Z(3)^0, 0*Z(3) ], [ 0*Z(3), Z(3) ] ] ), Group( [ [ 0*Z(3), Z(3)^0 ], [ Z(3), 0*Z(3) ] ], [ [ Z(3)^0, Z(3) ], [ Z(3), Z(3) ] ] ), Group( [ [ 0*Z(3), Z(3)^0 ], [ Z(3)^0, Z(3) ] ] ) ]
OneIrreducibleSolvableGroup( fun1, val1, fun2, val2, ... )
OneIrreducibleSolvableGroup
returns one irreducible solvable linear
group that has the properties given as arguments. Each property is
specified by passing a pair of arguments, the first being a function
which will be applied to all groups in the library, and the second being
a value or a list of values that this function must return in order to
have this group returned by OneIrreducibleSolvableGroup
. If no such
group exists, false
is returned.
gap> OneIrreducibleSolvableGroup( Dimension, 4, > IsLinearlyPrimitive, false ); Group( [ [ 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2) ], [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ], [ Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2) ], [ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ] ], [ [ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ], [ Z(2)^0, Z(2)^0, 0*Z(2), 0*Z(2) ], [ 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2) ], [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ] ], [ [ Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2) ], [ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ], [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ], [ 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0 ] ] )
AllIrreducibleSolvableGroups
and OneIrreducibleSolvableGroup
recognize the following functions and handle them very efficiently
(because the information is stored with the groups and so no computations
are needed): Dimension
for the linear degree, CharFFE
for the field
characteristic, Size
, IsLinearlyPrimitive
, and
MinimalBlockDimension
. Note that the last two are dummy functions that
can be used only in this context. Their meaning is explained at the end
of this section.
IrreducibleSolvableGroup( n, p, i )
IrreducibleSolvableGroup
returns the i-th irreducible solvable
subgroup of GL( n, p ). The irreducible solvable subgroups of
gap> g := IrreducibleSolvableGroup( 3, 5, 12 ); Group( [ [ 0*Z(5), Z(5)^2, 0*Z(5) ], [ Z(5)^2, 0*Z(5), 0*Z(5) ], [ 0*Z(5), 0*Z(5), Z(5)^2 ] ], [ [ 0*Z(5), Z(5)^0, 0*Z(5) ], [ 0*Z(5), 0*Z(5), Z(5)^0 ], [ Z(5)^0, 0*Z(5), 0*Z(5) ] ], [ [ Z(5)^2, 0*Z(5), 0*Z(5) ], [ 0*Z(5), Z(5)^0, 0*Z(5) ], [ 0*Z(5), 0*Z(5), Z(5)^2 ] ], [ [ Z(5)^0, 0*Z(5), 0*Z(5) ], [ 0*Z(5), Z(5)^2, 0*Z(5) ], [ 0*Z(5), 0*Z(5), Z(5)^2 ] ], [ [ Z(5), 0*Z(5), 0*Z(5) ], [ 0*Z(5), Z(5), 0*Z(5) ], [ 0*Z(5), 0*Z(5), Z(5) ] ] )
Apart from the usual components described in Group Records, the group records returned by the above functions have the following components.
size
:
isLinearlyPrimitive
:false
if G preserves a direct sum decomposition of the
underlying vector space, and true
otherwise.
minimalBlockDimension
:This library was computed and brought into GAP format by Mark Short. Descriptions of the algorithms used in computing the library data can be found in Sho92.
The GAP library of finite perfect groups provides, up to isomorphism, a list of all perfect groups whose sizes are less than 10^6 excluding the following: item For n = 61440, 122880, 172032, 245760, 344064, 491520, 688128, or 983040, the perfect groups of size n have not completely been determined yet. The library neither provides the number of these groups nor the groups themselves. vspace-1mm item For n = 86016, 368640, or 737280, the library does not yet contain the perfect groups of size n, it only provides their their numbers which are 52, 46, or 54, respectively. vspace-2mm
Except for these eleven sizes, the list of altogether 1096 perfect groups in the library is complete. It relies on results of Derek F.~Holt and Wilhelm Plesken which are published in their book it Perfect Groups HP89. Moreover, they have supplied to us files with presentations of 488 of the groups. In terms of these, the remaining 607 nontrivial groups in the library can be described as 276 direct products, 107 central products, and 224 subdirect products. They are computed automatically by suitable GAP functions whenever they are needed.
We are grateful to Derek Holt and Wilhelm Plesken for making their groups available to the GAP community by contributing their files. It should be noted that their book contains a lot of further information for many of the library groups. So we would like to recommend it to any GAP user who is interested in the groups.
The library has been brought into GAP format by Volkmar Felsch.
Like most of the other GAP libraries, the library of finite perfect
groups provides an extraction function, PerfectGroup
. It returns the
specified group in form of a finitely presented group which, in its group
record, bears some additional information that allows you to easily
construct an isomorphic permutation group of some appropriate degree by
just calling the PermGroup
function.
Further, there is a function NumberPerfectGroups
which returns the
number of perfect groups of a given size.
The library does not provide a selection or an example function. There
is, however, a function DisplayInformationPerfectGroups
which allows
the display of some information about arbitrary library groups without
actually loading the large files with their presentations, and without
constructing the groups themselves.
Moereover, there are two functions which allow you to formulate loops
over selected library groups. The first one is the
NumberPerfectLibraryGroups
function which, for any given size, returns
the number of groups in the library which are of that size.
The second one is the SizeNumbersPerfectGroups
function. It allows you
to ask for all library groups which contain certain composition factors.
The answer is provided in form of a list of pairs [size,n] where each
such pair characterizes the n^{rm th} library group of size size.
We will call such a pair [size,n] the size number of the respective
perfect group. As the size numbers are accepted as input arguments by the
PerfectGroup
and the DisplayInformationPerfectGroups
function, you
may use their list to formulate a loop over the associated groups.
Now we shall give an individual description of each library function.
NumberPerfectGroups( size )
NumberPerfectGroups
returns the number of non-isomorphic perfect groups
of size size for each positive integer size up to 10^6 except for
the eight sizes listed at the beginning of this section for which the
number is not yet known. For these values as well as for any argument out
of range it returns the value -1.
NumberPerfectLibraryGroups( size )
NumberPerfectLibraryGroups
returns the number of perfect groups of size
size which are available in the library of finite perfect groups.
The purpose of the function is to provide a simple way to formulate a loop over all library groups of a given size.
SizeNumbersPerfectGroups( factor1, factor2 ... )
SizeNumbersPerfectGroups
returns a list of the it size numbers (see
above) of all library groups that contain the specified factors among
their composition factors. Each argument must either be the name of a
simple group or an integer expression which is the product of the sizes
of one or more cyclic factors. The function ignores the order in which
the argmuments are given and, in fact, replaces any list of more than one
integer expression among the arguments by their product.
The following text strings are accepted as simple group names.
item[] "A5"
, "A6"
, "A7"
, "A8"
, "A9"
or "A(5)"
, "A(6)"
,
"A(7)"
, "A(8)"
, "A(9)"
for the alternating groups A_n, 5
leq n leq 9, vspace-2mm
item[] "L2(
q)"
or "L(2,
q)"
for PSL(2,q), where q is any
prime power with 4 leq q leq 125, vspace-2mm
item[] "L3(
q)"
or "L(3,
q)"
for PSL(3,q) with 2 leq q
leq 5, vspace-2mm
item[] "U3(
q)"
or "U(3,
q)"
for PSU(2,q) with 3 leq q
leq 5, vspace-2mm
item[] "U4(2)
or "U(4,2)"
for PSU(4,2), vspace-2mm
item[] "Sp4(4)"
or "S(4,4)"
for the symplectic group S(4,4),
vspace-2mm
item[] "Sz(8)"
for the Suzuki group Sz(8), vspace-2mm
item[] "M11"
, "M12"
, "M22"
or "M(11)"
, "M(12)"
, "M(22)"
for
the Matthieu groups M_{11}, M_{12}, and M_{22}, and
vspace-2mm
item[] "J1"
, "J2"
or "J(1)"
, "J(2)"
for the Janko groups J_1
and J_2. vspace-2mm
Note that, for most of the groups, the preceding list offers two
different names in order to be consistent with the notation used in
HP89 as well as with the notation used in the
DisplayCompositionSeries
command of GAP. However, as the names are
compared as text strings, you are restricted to the above choice. Even
expressions like "L2(
~32
~)"
or "L2(2^5)"
are not accepted.
As the use of the term PSU(n,q) is not unique in the literature, we state that here it denotes the factor group of SU(n,q) by its centre, where SU(n,q) is the group of all n times n unitary matrices with entries in GF(q^2) and determinant 1.
The purpose of the function is to provide a simple way to formulate a loop over all library groups which contain certain composition factors.
DisplayInformationPerfectGroups( size )
DisplayInformationPerfectGroups( size, n )
DisplayInformationPerfectGroups( [ size, n ] )
DisplayInformationPerfectGroups
displays some information about the
library group G, say, which is specified by the size number
[size,n] or by the two arguments size and n. If, in the second
case, n is omitted, the function will loop over all library groups of
size size.
The information provided for G includes the following items: item a headline containing the size number [size,n] of G in the form size.n (the suffix .n will be suppressed if, up to isomorphism, G is the only perfect group of size size), vspace-2mm item a message if G is simple or quasisimple, i.,e., if the factor group of G by its centre is simple, vspace-2mm item the ``description'' of the structure of G as it is given by Holt and Plesken in HP89 (see below), vspace-2mm item the size of the centre of G (suppressed, if G is simple), vspace-2mm item the prime decomposition of the size of G, vspace-2mm item orbit sizes for a faithful permutation representation of G which is provided by the library (see below), vspace-2mm item a reference to each occurrence of G in the tables of section 5.3 of HP89. Each of these references consists of a class number and an internal number (i,j) under which G is listed in that class. For some groups, there is more than one reference because these groups belong to more than one of the classes in the book. vspace-2mm Example:
gap> DisplayInformationPerfectGroups( 30720, 3 ); #I Perfect group 30720.3: A5 ( 2^4 E N 2^1 E 2^4 ) A #I centre = 1 size = 2^11*3*5 orbit size = 240 #I Holt-Plesken class 1 (9,3) gap> DisplayInformationPerfectGroups( 30720, 6 ); #I Perfect group 30720.6: A5 ( 2^4 x 2^4 ) C N 2^1 #I centre = 2 size = 2^11*3*5 orbit size = 384 #I Holt-Plesken class 1 (9,6) gap> DisplayInformationPerfectGroups( Factorial( 8 ) / 2 ); #I Perfect group 20160.1: A5 x L3(2) 2^1 #I centre = 2 size = 2^6*3^2*5*7 orbit sizes = 5 + 16 #I Holt-Plesken class 31 (1,1) (occurs also in class 32) #I Perfect group 20160.2: A5 2^1 x L3(2) #I centre = 2 size = 2^6*3^2*5*7 orbit sizes = 7 + 24 #I Holt-Plesken class 31 (1,2) (occurs also in class 32) #I Perfect group 20160.3: ( A5 x L3(2) ) 2^1 #I centre = 2 size = 2^6*3^2*5*7 orbit size = 192 #I Holt-Plesken class 31 (1,3) #I Perfect group 20160.4: simple group A8 #I size = 2^6*3^2*5*7 orbit size = 8 #I Holt-Plesken class 26 (0,1) #I Perfect group 20160.5: simple group L3(4) #I size = 2^6*3^2*5*7 orbit size = 21 #I Holt-Plesken class 27 (0,1)
For any library group G, the library files do not only provide a
presentation, but, in addition, a list of one or more subgroups S_1,
ldots, S_r of G such that there is a faithful permutation
representation of G of degree sum_{i=1}^{r} G !:! S_i on the set
{ S_i g mid 1 leq i leq r, , g in G } of the cosets of the
S_i. The respective permutation group is available via the PermGroup
function described below. The DisplayInformationPerfectGroups
function
displays only the available degree. The message
orbit size = 8
in the above example means that the available permutation representation is transitive and of degree 8, whereas the message
orbit sizes = 7 + 24
means that a nontransitive permutation representation is available which acts on two orbits of size 7 and 24 respectively.
The notation used in the ``description'' of a group is explained in section 5.1.2 of HP89. We quote the respective page from there:
sl `Within a class Q,#,p, an isomorphism type of groups will be denoted by an ordered pair of integers (r,n), where r geq 0 and n > 0. More precisely, the isomorphism types in Q # p of order p^r !! mid !! Q !! mid will be denoted by (r,1), (r,2), (r,3), ldots,. Thus Q will always get the size number (0,1).
In addition to the symbol (r,n), the groups in Q , {sl #} , p will also be given a more descriptive name. The purpose of this is to provide a very rough idea of the structure of the group. The names are derived in the following manner. First of all, the isomorphism classes of irreducible F_pQ-modules M with mid !! Q !! mid mid !! M !! mid , leq 10^6, where F_p is the field of order p, are assigned symbols. These will either be simply p^x, where x is the dimension of the module, or, if there is more than one isomorphism class of irreducible modules having the same dimension, they will be denoted by p^x, p^{x^prime}, etc. The one-dimensional module with trivial Q-action will therefore be denoted by p^1. These symbols will be listed under the description of Q. The group name consists essentially of a list of the composition factors working from the top of the group downwards; hence it always starts with the name of Q itself. (This convention is the most convenient in our context, but it is different from that adopted in the ATLAS (Conway it et al.~1985), for example, where composition factors are listed in the reverse order. For example, we denote a group isomorphic to SL(2,5) by A_5 2^1 rather than 2 cdot A_5.)
Some other symbols are used in the name, in order to give some idea of the relationship between these composition factors, and splitting properties. We shall now list these additional symbols. item[times] between two factors denotes a direct product of F_pQ-modules or groups. vspace-2mm item[C] (for `commutator') between two factors means that the second lies in the commutator subgroup of the first. Similarly, a segment of the form (f_1 ! times ! f_2) {sl C} f_3 would mean that the factors f_1 and f_2 commute modulo f_3 and f_3 lies in [f_1,f_2]. vspace-2mm item[A] (for `abelian') between two factors indicates that the second is in the pth power (but not the commutator subgroup) of the first. `A' may also follow the factors, if bracketed. vspace-7mm item[E] (for `elementary abelian') between two factors indicates that together they generate an elementary abelian group (modulo subsequent factors), but that the resulting F_pQ-module extension does not split. vspace-2mm item[N] (for `nonsplit') before a factor indicates that Q (or possibly its covering group) splits down as far at this factor but not over the factor itself. So `Q f_1 {sl N} f_2' means that the normal subgroup f_1f_2 of the group has no complement but, modulo f_2, f_1, does have a complement. vspace-2mm
We must stress that this notation does not always succeed in being precise or even unambiguous, and the reader is free to ignore it if it does not seem helpful.'
If such a group description has been given in the book for G (and, in
fact, this is the case for most of the library groups), it is displayed
by the DisplayInformationPerfectGroups
function. Otherwise the function
provides a less explicit description of the (in these cases unique)
Holt-Plesken class to which G belongs, together with a serial number if
this is necessary to make it unique.
PerfectGroup( size )
PerfectGroup( size, n )
PerfectGroup( [ size, n ] )
PerfectGroup
is the essential extraction function of the library. It
returns a finitely presented group, G say, which is isomorphic to the
library group specified by the size number [size,n] or by the two
separate arguments size and n. In the second case, you may omit the
parameter n. Then the default value is n = 1.
gap> G := PerfectGroup( 6048 ); PerfectGroup(6048) gap> G.generators; [ a, b ] gap> G.relators; [ a^2, b^6, a*b*a*b*a*b*a*b*a*b*a*b*a*b, a*b^2*a*b^2*a*b^2*a*b^-2*a*b^-2*a*b^-2, a*b*a*b^-2*a*b*a*b^-2*a*b*a*b^-2*a*b*a*b^-1*a*b^-1 ] gap> G.size; 6048 gap> G.description; "U3(3)" gap> G.subgroups; [ Subgroup( PerfectGroup(6048), [ a, b*a*b*a*b*a*b^3 ] ) ]
The generators and relators of G coincide with those given in HP89.
Note that, besides the components that are usually initialized for any finitely presented group, the group record of G contains the following components:
size
:
isPerfect
:true
,
description
:DisplayInformationPerfectGroups
function above,
source
:
subgroups
:The last of these components exists only if G is one of the 488 nontrivial library groups which are given directly by a presentation on file, i.,e., which are not constructed from other library groups in form of a direct, central, or subdirect product. It will be required by the following function.
PermGroup( G )
PermGroup
returns a permutation group, P say, which is isomorphic to
the given group G which is assumed to be a finitely presented perfect
group that has been extracted from the library of finite perfect groups
via the PerfectGroup
function.
Let S_1, ldots, S_r be the subgroups listed in the component
G.subgroups
of the group record of G. Then the resulting group P
is the permutation group of degree sum_{i=1}^{r} G !:! S_i which is
induced by G on the set { S_i g mid 1 leq i leq r, g in G } of
all cosets of the S_i.
Example (continued):
gap> P := PermGroup( G ); PermGroup(PerfectGroup(6048)) gap> P.size; 6048 gap> P.degree; 28
Note that some of the library groups do not have a faithful permutation representation of small degree. Computations in these groups may be rather time consuming.
Example:
gap> P := PermGroup( PerfectGroup( 129024, 2 ) ); PermGroup(PerfectGroup(129024,2)) gap> P.degree; 14336
A library of irreducible maximal finite integral matrix groups is provided with GAP. It contains Q-class representatives for all of these groups of dimension at most 24, and Z-class representatives for those of dimension at most 11 or of dimension 13, 17, 19, or 23.
The groups provided in this library have been determined by Wilhelm Plesken, partially as joint work with Michael Pohst, or by members of his institute (Lehrstuhl B faccent127ur Mathematik, RWTH Aachen). In particular, the data for the groups of dimensions 2 to 9 have been taken from the output of computer calculations which they performed in 1979 (see PP77, PP80). The Z-class representatives of the groups of dimension 10 have been determined and computed by Bernd Souvignier (Sou94), and those of dimensions 11, 13, and 17 have been recomputed for this library from the circulant Gram matrices given in Ple85, using the stand-alone programs for the computation of short vectors and Bravais groups which have been developed in Plesken's institute. The Z-class representatives of the groups of dimensions 19 and 23 had already been determined in Ple85. Gabriele Nebe has recomputed them for us. Her main contribution to this library, however, is that she has determined and computed the Q-class representatives of the groups of non-prime dimensions between 12 and 24 (see PN95, NP95, Neb95).
The library has been brought into GAP format by Volkmar Felsch. He has applied several GAP routines to check certain consistency of the data. However, the credit and responsibility for the lists remain with the authors. We are grateful to Wilhelm Plesken, Gabriele Nebe, and Bernd Souvignier for supplying their results to GAP.
In the preceding acknowledgement, we used some notations that will also be needed in the sequel. We first define these.
Any integral matrix group G of dimension n is a subgroup of GL_n(Z) as well as of GL_n(Q) and hence lies in some conjugacy class of integral matrix groups under GL_n(Z) and also in some conjugacy class of rational matrix groups under GL_n(Q). As usual, we call these classes the Z-class and the Q-class of G, respectively. Note that any conjugacy class of subgroups of GL_n(Q) contains at least one Z-class of subgroups of GL_n(Z) and hence can be considered as the Q-class of some integral matrix group.
In the context of this library we are only concerned with Z-classes and Q-classes of subgroups of GL_n(Z) which are irreducible and maximal finite in GL_n(Z) (we will call them em i.,m.,f.~subgroups of GL_n(Z)). We can distinguish two types of these groups:
First, there are those i.,m.,f.~subgroups of GL_n(Z) which are also maximal finite subgroups of GL_n(Q). Let us denote the set of their Q-classes by Q_1(n). It is clear from the above remark that Q_1(n) just consists of the Q-classes of, i.,m.,f.~subgroups of GL_n(Q).
Secondly, there is the set Q_2(n) of the Q-classes of the remaining i.,m.,f.~subgroups of GL_n(Z), i.,e., of those which are not maximal finite subgroups of GL_n(Q). For any such group G, say, there is at least one class C in Q_1(n) such that G is conjugate under Q to a proper subgroup of some group H in C. In fact, the class C is uniquely determined for any group G occurring in the library (though there seems to be no reason to assume that this property should hold in general). Hence we may call C the em rational i.,m.,f.~class of G. Finally, we will denote the number of classes in Q_1(n) and Q_2(n) by q_1(n) and q_2(n), respectively.
As an example, let us consider the case n = 4. There are 6 Z-classes of, i.,m.,f.~subgroups of GL_4(Z) with representative subgroups G_1, ldots, G_6 of isomorphsim types, mboxG_1 cong W(F_4),, mboxG_2 cong D_{12} wr C_2,, mboxG_3 cong G_4 cong C_2 times S_5,, mboxG_5 cong W(B_4),, and, mboxG_6 cong (D_{12} {sf Y} D_{12}) !:! C_2. The corresponding Q-classes, R_1, ldots, R_6, say, are pairwise different except that R_3 coincides with R_4. The groups G_1, G_2, and G_3 are i.,m.,f.~subgroups of GL_4(Q), but G_5 and G_6 are not because they are conjugate under GL_4(Q) to proper subgroups of G_1 and G_2, respectively. So we have Q_1(4) = { R_1, R_2, R_3 },, Q_2(4) = { R_5, R_6 },, q_1(4) = 3, and q_2(4) = 2.
The q_1(n) Q-classes of, i.,m.,f.~subgroups of GL_n(Q) have been determined for each dimension n leq 24. The current GAP library provides integral representative groups for all these classes. Moreover, all Z-classes of, i.,m.,f.~subgroups of GL_n(Z) are known for n leq 11 and for n in {13,17,19,23}. For these dimensions, the library offers integral representative groups for all Q-classes in Q_1(n) and Q_2(n) as well as for all Z-classes of, i.,m.,f.~subgroups of GL_n(Z).
Any group G of dimension n given in the library is represented as the automorphism group G = {rm Aut}(F,L) = { g in GL_n(Z) mid Lg = L ; {rm and} ; g F g^{rm tr} = F } of a positive definite symmetric n times n matrix F in Z^{n times n} on an n-dimensional lattice L cong Z^{1 times n} (for details see e.,g. PN95). GAP provides for G a list of matrix generators and the em Gram matrix F.
The positive definite quadratic form defined by F defines a em norm v F v^{rm tr} for each vector v in L, and there is only a finite set of vectors of minimal norm. These vectors are often simply called the em ``short vectors''. Their set splits into orbits under G, and G being irreducible acts faithfully on each of these orbits by multiplication from the right. GAP provides for each of these orbits the orbit size and a representative vector.
Like most of the other GAP libraries, the library of,
i.,m.,f.~integral matrix groups supplies an extraction function,
ImfMatGroup
. However, as the library involves only 423 different
groups, there is no need for a selection or an example function. Instead,
there are two functions, ImfInvariants
and DisplayImfInvariants
,
which provide some Z-class invariants that can be extracted from the
library without actually constructing the representative groups
themselves. The difference between these two functions is that the latter
one displays the resulting data in some easily readable format, whereas
the first one returns them as record components so that you can properly
access them.
We shall give an individual description of each of the library functions,
but first we would like to insert a short remark concerning their names:
Any self-explaining name of a function handling em irreducible
maximal finite integral matrix groups would have to include this term in
full length and hence would grow extremely long. Therefore we have
decided to use the abbreviation Imf
instead in order to restrict the
names to some reasonable length.
The first three functions can be used to formulate loops over the classes.
ImfNumberQQClasses( dim )
ImfNumberQClasses( dim )
ImfNumberZClasses( dim, q )
ImfNumberQQClasses
returns the number q_1(dim) of Q-classes
of, i.,m.,f.~rational matrix groups of dimension dim. Valid values
of dim are all positive integers up to 24.
Note: In order to enable you to loop just over the classes belonging to Q_1(dim), we have arranged the list of Q-classes of dimension dim for any dimension dim in the library such that, whenever the classes of Q_2(dim) are known, too, i.,e., in the cases dim leq 11 or dim in {13,17,19,23}, the classes of Q_1(dim) precede those of Q_2(dim) and hence are numbered from 1 to q_1(dim).
ImfNumberQClasses
returns the number of Q-classes of groups of
dimension dim which are available in the library. If dim leq 11 or
dim in {13,17,19,23}, this is the number q_1(dim) +
q_2(dim) of Q-classes of, i.,m.,f.~subgroups of
GL_{dim}(Z). Otherwise, it is just the number q_1(dim) of
Q-classes of, i.,m.,f.~subgroups of GL_{dim}(Q). Valid values of
dim are all positive integers up to 24.
ImfNumberZClasses
returns the number of Z-classes in the q^{rm
th} Q-class of, i.,m.,f.~integral matrix groups of dimension
dim. Valid values of dim are all positive integers up to 11 and all
primes up to 23.
DisplayImfInvariants( dim, q )
DisplayImfInvariants( dim, q, z )
DisplayImfInvariants
displays the following Z-class invariants of
the groups in the z^{rm th} Z-class in the q^{rm th}
Q-class of i.,m.,f.~integral matrix groups of dimension dim:
item its Z-class number in the form dim.q.z, if dim is at
most 11 or a prime, or its Q-class number in the form dim.q,
else, vspace-2mm
item a message if the group is solvable, vspace-2mm
item the size of the group, vspace-2mm
item the isomorphism type of the group, vspace-2mm
item the elementary divisors of the associated quadratic form,
vspace-2mm
item the sizes of the orbits of short vectors (these sizes are the
degrees of the faithful permutation representations which you may
construct using the PermGroup
or PermGroupImfGroup
commands
below), vspace-2mm
item the norm of the associated short vectors, vspace-2mm
item only in case that the group is not an i.,m.,f.~group in
GL_n(Q): an appropriate message, including the Q-class
number of the corresponding rational i.,m.,f.~class.
vspace-2mm
If you specify the value 0 for any of the parameters dim, q, or z,
the command will loop over all available dimensions, Q-classes of
given dimension, or Z-classes within the given Q-class,
respectively. Otherwise, the values of the arguments must be in range. A
value z neq 1 must not be specified if the Z-classes are not
known for the given dimension, i.,e., if dim > 11 and dim not in
{13,17,19,23}. The default value of z is~1. This value of z will
be accepted even if the Z-classes are not known. Then it specifies the
only representative group which is available for the q^{rm th}
Q-class. The greatest legal value of dim is 24.
gap> DisplayImfInvariants( 3, 1, 0 ); #I Z-class 3.1.1: Solvable, size = 2^4*3 #I isomorphism type = C2 wr S3 = C2 x S4 = W(B3) #I elementary divisors = 1^3 #I orbit size = 6, minimal norm = 1 #I Z-class 3.1.2: Solvable, size = 2^4*3 #I isomorphism type = C2 wr S3 = C2 x S4 = C2 x W(A3) #I elementary divisors = 1*4^2 #I orbit size = 8, minimal norm = 3 #I Z-class 3.1.3: Solvable, size = 2^4*3 #I isomorphism type = C2 wr S3 = C2 x S4 = C2 x W(A3) #I elementary divisors = 1^2*4 #I orbit size = 12, minimal norm = 2 gap> DisplayImfInvariants( 8, 15, 1 ); #I Z-class 8.15.1: Solvable, size = 2^5*3^4 #I isomorphism type = C2 x (S3 wr S3) #I elementary divisors = 1*3^3*9^3*27 #I orbit size = 54, minimal norm = 8 #I not maximal finite in GL(8,Q), rational imf class is 8.5 gap> DisplayImfInvariants( 20, 23 ); #I Q-class 20.23: Size = 2^5*3^2*5*11 #I isomorphism type = (PSL(2,11) x D12).C2 #I elementary divisors = 1^18*11^2 #I orbit size = 3*660 + 2*1980 + 2640 + 3960, minimal norm = 4
Note that the DisplayImfInvariants
function uses a kind of shorthand to
display the elementary divisors. E.~g., the expression 1*3^3*9^3*27
in
the preceding example stands for the elementary divisors
1,3,3,3,9,9,9,27. (See also the next example which shows that the
ImfInvariants
function provides the elementary divisors in form of an
ordinary GAP list.)
In the description of the isomorphism types the following notations are
used:
item[]
item[]
item[makebox[15mm][l]A,mbox{x},B] denotes a direct product of
a group A by a group B, vspace-1mm
item[makebox[15mm][l]A,mbox{subd},B] denotes a subdirect
product of A by B, vspace-1mm
item[makebox[15mm][l]A,mbox{Y},B] denotes a central product of
A by B, vspace-1mm
item[makebox[15mm][l]A,mbox{wr},B] denotes a wreath product of
A by B, vspace-1mm
item[makebox[15mm][l]A:B] denotes a split extension of A by
B, vspace-1mm
item[makebox[15mm][l]A,.,B] denotes just an extension of A by
B (split or nonsplit). vspace-2mm
The groups involved are
item the cyclic groups C_n, dihedral groups D_n, and generalized
quaternion groups Q_n of order n, denoted by C
n, D
n,
and Q
n, respectively, vspace-2mm
item the alternating groups A_n and symmetric groups S_n of degree
n, denoted by A
n and S
n, respectively, vspace-2mm
item the linear groups GL_n(q), PGL_n(q), SL_n(q), and PSL_n(q),
denoted by GL(
n,
q)
, PGL(
n,
q)
,
SL(
n,
q)
, and PSL(
n,
q)
, respectively,
vspace-2mm
item the unitary groups SU_n(q) and PSU_n(q), denoted by
SU(
n,
q)
and PSU(
n,
q)
, respectively,
vspace-2mm
item the symplectic groups Sp(n,q), denoted by Sp(
n,
q)
,
vspace-2mm
item the orthogonal group O_8^{,+}(2), denoted by O+(8,2)
,
vspace-2mm
item the extraspecial groups 2_+^{,1+8}, 3_+^{,1+2},
3_+^{,1+4}, and 5_+^{,1+2}, denoted by 2+^(1+8)
,
3+^(1+2)
, 3+^(1+4)
, and 5+^(1+2)
, respectively, vspace-2mm
item the Chevalley group G_2(3), denoted by G(2,3)
,
vspace-2mm
item the Weyl groups W(A_n), W(B_n), W(D_n), W(E_n), and
W(F_4), denoted by W(A
n)
, W(B
n)
, W(D
n)
,
W(E
n)
, and W(F4)
, respectively, vspace-2mm
item the sporadic simple groups Co_1, Co_2, Co_3, HS, J_2,
M_{12}, M_{22}, M_{23}, M_{24}, and Mc, denoted by
Co1
, Co2
, Co3
, HS
, J2
, M12
, M22
, M23
, M24
, and
Mc
, respectively, vspace-2mm
item a point stabilizer of index 11 in M_{11}, denoted by M10
.
vspace-2mm
As mentioned above, the data assembled by the DisplayImfInvariants
command are ``cheap data'' in the sense that they can be provided by
the library without loading any of its large matrix files or performing
any matrix calculations. The following function allows you to get proper
access to these cheap data instead of just displaying them.
ImfInvariants( dim, q )
ImfInvariants( dim, q, z )
ImfInvariants
returns a record which provides some Z-class
invariants of the groups in the z^{rm th} Z-class in the
q^{rm th} Q-class of i.,m.,f.~integral matrix groups of
dimension dim. A value z neq 1 must not be specified if the
Z-classes are not known for the given dimension, i.,e., if dim > 11
and dim not in {13,17,19,23}. The default value of z is~1. This
value of z will be accepted even if the Z-classes are not
known. Then it specifies the only representative group which is available
for the q^{rm th} Q-class. The greatest legal value of dim is
24.
The resulting record contains six or seven components:
size
:
isSolvable
:true
if G is solvable,
isomorphismType
:DisplayImfInvariants
command above),
elementaryDivisors
:ElementaryDivisorsMat
function,
see ElementaryDivisorsMat),
minimalNorm
:
sizesOrbitsShortVectors
:
maximalQClass
:Note that four of these data, namely the group size, the solvability, the isomorphism type, and the corresponding rational i.,m.,f.~class, are not only Z-class invariants, but also Q-class invariants.
Note further that, though the isomorphism type is a Q-class invariant, you will sometimes get different descriptions for different Z-classes of the same Q-class (as, e.,g., for the classes 3.1.1 and 3.1.2 in the last example above). The purpose of this behaviour is to provide some more information about the underlying lattices.
gap> ImfInvariants( 8, 15, 1 ); rec( size := 2592, isSolvable := true, isomorphismType := "C2 x (S3 wr S3)", elementaryDivisors := [ 1, 3, 3, 3, 9, 9, 9, 27 ], minimalNorm := 8, sizesOrbitsShortVectors := [ 54 ], maximalQClass := 5 ) gap> ImfInvariants( 24, 1 ).size; 10409396852733332453861621760000 gap> ImfInvariants( 23, 5, 2 ).sizesOrbitsShortVectors; [ 552, 53130 ] gap> for i in [ 1 .. ImfNumberQClasses( 22 ) ] do > Print( ImfInvariants( 22, i ).isomorphismType, "\n" ); od; C2 wr S22 = W(B22) (C2 x PSU(6,2)).S3 (C2 x S3) wr S11 = (C2 x W(A2)) wr S11 (C2 x S12) wr C2 = (C2 x W(A11)) wr C2 C2 x S3 x S12 = C2 x W(A2) x W(A11) (C2 x HS).C2 (C2 x Mc).C2 C2 x S23 = C2 x W(A22) C2 x PSL(2,23) C2 x PSL(2,23) C2 x PGL(2,23) C2 x PGL(2,23)
ImfMatGroup( dim, q )
ImfMatGroup( dim, q, z )
ImfMatGroup
is the essential extraction function of this library. It
returns a representative group, G say, of the z^{rm th} Z-class
in the q^{rm th} Q-class of i.,m.,f.~integral matrix groups of
dimension dim. A value z neq 1 must not be specified if the
Z-classes are not known for the given dimension, i.,e., if dim > 11
and dim not in {13,17,19,23}. The default value of z is~1. This
value of z will be accepted even if the Z-classes are not
known. Then it specifies the only representative group which is available
for the q^{rm th} Q-class. The greatest legal value of dim is
24.
gap> G := ImfMatGroup( 5, 1, 3 ); ImfMatGroup(5,1,3) gap> for m in G.generators do PrintArray( m ); od; [ [ -1, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0 ], [ 0, 0, 0, 1, 0 ], [ -1, -1, -1, -1, 2 ], [ -1, 0, 0, 0, 1 ] ] [ [ 0, 1, 0, 0, 0 ], [ 0, 0, 1, 0, 0 ], [ 0, 0, 0, 1, 0 ], [ 1, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 1 ] ]
The group record of G contains the usual components of a matrix group
record. In addition, it includes the same six or seven records as the
resulting record of the ImfInvariants
function described above, namely
the components size
, isSolvable
, isomorphismType
,
elementaryDivisors
, minimalNorm
, and sizesOrbitsShortVectors
and,
if G is not a rational i.,m.,f.~group, maximalQClass
. Moreover,
there are the two components
form
:
repsOrbitsShortVectors
:
The last of these components will be required by the PermGroup
function
below.
Example:
gap> G.size; 3840 gap> G.isomorphismType; "C2 wr S5 = C2 x W(D5)" gap> PrintArray( G.form ); [ [ 4, 0, 0, 0, 2 ], [ 0, 4, 0, 0, 2 ], [ 0, 0, 4, 0, 2 ], [ 0, 0, 0, 4, 2 ], [ 2, 2, 2, 2, 5 ] ] gap> G.elementaryDivisors; [ 1, 4, 4, 4, 4 ] gap> G.minimalNorm; 4
If you want to perform calculations in such a matrix group G you should be aware of the fact that GAP offers much more efficient permutation group routines than matrix group routines. Hence we recommend that you do your computations, whenever it is possible, in the isomorphic permutation group that is induced by the action of G on one of the orbits of the associated short vectors. You may call one of the following functions to get such a permutation group.
PermGroup( G )
PermGroup
returns the permutation group which is induced by the given
i.,m.,f.~integral matrix group G on an orbit of minimal size of G
on the set of short vectors (see also PermGroupImfGroup
below).
The permutation representation is computed by first constructing the specified orbit, S say, of short vectors and then computing the permutations which are induced on S by the generators of G. We would like to warn you that in case of a large orbit this procedure may be space and time consuming. Fortunately, there are only five Q-classes in the library for which the smallest orbit of short vectors is of size greater than 20000, the worst case being the orbit of size 196560 for the Leech lattice (dim = 24, q = 3).
As mentioned above, PermGroup
constructs the required permutation
group, P say, as the image of G under the isomorphism between the
matrices in G and their action on S. Moreover, it constructs the
inverse isomorphism from P to G, varphi say, and returns it in the
group record component P.bijection
of P. In fact, varphi is
constructed by determining a Q-base B subset S of Q^{1 times
dim} in S and, in addition, the associated base change matrix M
which transforms B into the standard base of Z^{1 times dim}. Then
the image varphi(p) of any permutation p in P can be easily
computed: If, for 1 leq i leq dim, b_i is the position number in
S of the i^{rm th} base vector in B, it suffices to look up the
vector whose position number in S is the image of b_i under p and
to multiply this vector by M to get the i^{rm th} row of
varphi(p).
You may use varphi at any time to compute the images in G of permutations in P or to compute the preimages in P of matrices in G.
The record of P contains, in addition to the usual components of permutation group records, some special components. The most important of those are:
isomorphismType
:DisplayImfInvariants
command above),
matGroup
:
bijection
:
orbitShortVectors
:
baseVectorPositions
:
baseChangeMatrix
:As an example, let us compute a set R of matrix generators for the solvable residuum of the group G that we have constructed in the preceding example.
gap> # Perform the computations in an isomorphic permutation group. gap> P := PermGroup( G ); PermGroup(ImfMatGroup(5,1,3)) gap> P.generators; [ ( 1, 7, 6)( 2, 9)( 4, 5,10), ( 2, 3, 4, 5)( 6, 9, 8, 7) ] gap> D := DerivedSubgroup( P ); Subgroup( PermGroup(ImfMatGroup(5,1,3)), [ ( 1, 2,10, 9)( 3, 8)( 4, 5)( 6, 7), ( 1, 6)( 2, 7, 9, 4)( 3, 8)( 5,10), ( 1, 5,10, 6)( 2, 8, 9, 3) ] ) gap> Size( D ); 960 gap> IsPerfect( D ); true gap> # Now move the results back to the matrix group. gap> phi := P.bijection;; gap> R := List( D.generators, p -> Image( phi, p ) );; gap> for m in R do PrintArray( m ); od; [ [ -1, -1, -1, -1, 2 ], [ 0, -1, 0, 0, 0 ], [ 0, 0, 0, 1, 0 ], [ 0, 0, 1, 0, 0 ], [ -1, -1, 0, 0, 1 ] ] [ [ 0, 0, -1, 0, 0 ], [ 0, -1, 0, 0, 0 ], [ 1, 0, 0, 0, 0 ], [ -1, -1, -1, -1, 2 ], [ 0, -1, -1, 0, 1 ] ] [ [ 0, -1, 0, 0, 0 ], [ 1, 0, 0, 0, 0 ], [ 0, 0, 1, 0, 0 ], [ -1, -1, -1, -1, 2 ], [ 0, -1, 0, -1, 1 ] ] gap> # The PreImage function allows us to use the inverse of phi. gap> PreImage( phi, R[1] ) = D.generators[1]; true
PermGroupImfGroup( G, n )
PermGroupImfGroup
returns the permutation group which is induced by the
given i.,m.,f.~integral matrix group G on its n^{rm th} orbit of
short vectors. The only difference to the above PermGroup
function is
that you can specify the orbit to be used. In fact, as the orbits of
short vectors are sorted by increasing sizes, the function PermGroup(
G )
has been implemented such that it is equivalent to
PermGroupImfGroup( G, 1 )
.
gap> ImfInvariants( 12, 9 ).sizesOrbitsShortVectors; [ 120, 300 ] gap> G := ImfMatGroup( 12, 9 ); ImfMatGroup(12,9) gap> P1 := PermGroupImfGroup( G, 1 ); PermGroup(ImfMatGroup(12,9)) gap> P1.degree; 120 gap> P2 := PermGroupImfGroup( G, 2 ); PermGroupImfGroup(ImfMatGroup(12,9),2) gap> P2.degree; 300
GAP provides a library of crystallographic groups of dimensions 2, 3, and 4 which covers many of the data that have been listed in the book ``Crystallographic groups of four-dimensional space'' BBNWZ78. It has been brought into GAP format by Volkmar Felsch.
How to access the data of the book
Among others, the library offers functions which provide access to the
data listed in the Tables 1, 5, and 6 of BBNWZ78:
item The information on the crystal families listed in Table 1 can be
reproduced using the DisplayCrystalFamily
function.
item Similarly, the DisplayCrystalSystem
function can be used to
reproduce the information on the crystal systems provided in Table
1.
item The information given in the Q-class headlines of Table 1 can
be displayed by the DisplayQClass
function, whereas the
FpGroupQClass
function can be used to reproduce the presentations
that are listed in Table 1 for the Q-class representatives.
item The information given in the Z-class headlines of Table 1 will
be covered by the results of the DisplayZClass
function, and the
matrix generators of the Z-class representatives can be
constructed by calling the MatGroupZClass
function.
item The DisplaySpaceGroupType
and the DisplaySpaceGroupGenerators
functions can be used to reproduce all of the information on the
space-group types that is provided in Table 1.
item The normalizers listed in Table 5 can be reproduced by calling the
NormalizerZClass
function.
item Finally, the CharTableQClass
function will compute the character
tables listed in Table 6, whereas the isomorphism types given in
Table 6 may be obtained by calling the DisplayQClass
function.
vspace-2mm
The display functions mentioned in the above list print their output with
different indentation. So, calling them in a suitably nested loop, you
may produce a listing in which the information about the objects of
different type will be properly indented as has been done in Table 1 of
BBNWZ78.
Representation of space groups in GAP
Probably the most important function in the library is the SpaceGroup
function which provides representatives of the affine classes of space
groups. A space group of dimension n is represented by an
(n+1)-dimensional rational matrix group as follows.
If S is an n-dimensional space group, then each element alpha in S is an affine mapping alpha!: V rightarrow V of an n-dimensional R-vector space V onto itself. Hence alpha can be written as the sum of an appropriate invertible linear mapping varphi!: V rightarrow V and a translation by some translation vector t in V such that, if we write mappings from the left, we have alpha(v) = varphi(v) + t for all v in V.
If we fix a basis of V and then replace each v in V by the column vector of its coefficients with respect to that basis (and hence V by the isomorphic column vector space R^{n times 1}), we can describe the linear mapping varphi involved in alpha by an n times n matrix M_varphi in GL_n(R) which acts by multiplication from the left on the column vectors in R^{n times 1}. Hence, if we identify V with R^{n times 1}, we have alpha(v) = M_varphi cdot v + t for all v in R^{n times 1}.
Moreover, if we extend each column vector v in R^{n times 1} to a column left[ ! ! catcode`|=12 begin{tabular}{c} catcode`|=13 v \ 1 end{tabular} ! ! right] of length n+1 by adding an entry 1 in the last position and if we define an (n+1) times (n+1) matrix M_alpha = left[ catcode`|=12 begin{tabular}{c|c} catcode`|=13 M_varphi & t \ hline 0 & 1 end{tabular} right], we have left[ ! ! catcode`|=12 begin{tabular}{c} catcode`|=13 alpha(v) \ 1 for all v in R^n times 1. This means that we can represent the space group S by the isomorphic group M(S) = { M_alpha mid alpha in S }. The submatrices M_varphi occurring in the elements of M(S) form an n times n matrix group P(S), the ``point group'' of M(S). In fact, we can choose the basis of R^n times 1 such that M_varphi in GL_n(Z) and t in Q^n times 1 for all M_alpha in M(S). In particular, the space group representatives that are normally used by the crystallographers are of this form, and the book BBNWZ78 uses the same convention.
Before we describe all available library functions in detail, we have to add three remarks.
Remark 1
The concepts used in this section are defined in chapter 1 (Basic definitions) of BBNWZ78. However, note that the definition of the concept of a crystal system given on page 16 of that book relies on the following statement about Q-classes: item[] For a Q-class C there is a unique holohedry H such that each f.,u. group in C is a subgroup of some f.,u. group in H, but is not a subgroup of any f.,u. group belonging to a holohedry of smaller order. vspace-2mm This statement is correct for dimensions 1, 2, 3, and 4, and hence the definition of ``crystal system'' given on page 16 of BBNWZ78 is known to be unambiguous for these dimensions. However, there is a counterexample to this statement in seven-dimensional space so that the definition breaks down for some higher dimensions.
Therefore, the authors of the book have since proposed to replace this definition of ``crystal system'' by the following much simpler one, which has been discussed in more detail in NPW81. To formulate it, we use the intersections of Q-classes and Bravais flocks as introduced on page 17 of BBNWZ78, and we define the classification of the set of all Z-classes into crystal systems as follows: item[] Definition: A crystal system (introduced as an equivalence class of Z-classes) consists of full geometric crystal classes. The Z-classes of two (geometric) crystal classes belong to the same crystal system if and only if these geometric crystal classes intersect the same set of Bravais flocks of Z-classes. vspace-2mm From this definition of a crystal system of Z-classes one then obtains crystal systems of f.,u. groups, of space-group types, and of space groups in the same manner as with the preceding definitions in the book.
The new definition is unambiguous for all dimensions. Moreover, it can be checked from the tables in the book that it defines the same classification as the old one for dimensions 1, 2, 3, and 4.
It should be noted that the concept of crystal family is well-defined independently of the dimension if one uses the ``more natural'' second definition of it at the end of page 17. Moreover, the first definition of crystal family on page 17 defines the same concept as the second one if the now proposed definition of crystal system is used.
Remark 2
The second remark just concerns a different terminology in the tables of BBNWZ78 and in the current library. In group theory, the number of elements of a finite group usually is called the ``order'' of the group. This notation has been used throughout in the book. Here, however, we will follow the GAP conventions and use the term ``size'' instead.
Remark 3
The third remark concerns a problem in the use of the space groups that should be well understood.
There is an alternative to the representation of the space group elements by matrices of the form left[ catcode`|=12 begin{tabular}{c|c} catcode`|=13 M_varphi & t \ hline 0 & 1 end{tabular} right] as described above. Instead of considering the coefficient vectors as columns we may consider them as rows. Then we can associate to each affine mapping alpha in S an (n+1) times (n+1) matrix overline{M}_alpha = left[ catcode`|=12 begin{tabular}{c|c} catcode`|=13 overlineM_overlinevarphi & 0 \ hline overlinet & 1 end{tabular} right] with overline{M}_{overline{varphi}} in GL_n(R) and overline{t} in R^{1 times n} such that [alpha(overline{v}),1] = [overline{v},1] cdot overline{M}_alpha for all overline{v} in R^{1 times n}, and we may represent S by the matrix group overline{M}(S) = { overline{M}_alpha mid alpha in S }. Again, we can choose the basis of R^{1 times n} such that overline{M}_{overline{varphi}} in GL_n(Z) and overline{t} in Q^{1 times n} for all overline{M}_alpha in overline{M}(S).
From the mathematical point of view, both approaches are equivalent. In particular, M(S) and overline{M}(S) are isomorphic, for instance via the isomorphism tau mapping M_alpha in M(S) to (M_alpha^{rm tr})^{-1}. Unfortunately, however, neither of the two is a good choice for our GAP library.
The first convention, using matrices which act on column vectors from the left, is not consistent with the fact that actions in GAP are usually from the right.
On the other hand, if we choose the second convention, we run into a problem with the names of the space groups as introduced in BBNWZ78. Any such name does not just describe the abstract isomorphism type of the respective space group S, but reflects properties of the matrix group M(S). In particular, it contains as a leading part the name of the Z-class of the associated point group P(S). Since the classification of space groups by affine equivalence is tantamount to their classification by abstract isomorphism, overline{M}(S) lies in the same affine class as M(S) and hence should get the same name as M(S). But the point group P(S) that occurs in that name is not always Z-equivalent to the point group overline{P}(S) of overline{M}(S). For example, the isomorphism tau!!: M(S) rightarrow overline{M}(S) defined above maps the Z-class representative with the parameters [3,7,3,2] (in the notation described below) to the Z-class representative with the parameters [3,7,3,3]. In other words: The space group names introduced for the groups M(S) in BBNWZ78 lead to confusing inconsistencies if assigned to the groups overline{M}(S).
In order to avoid this confusion we decided that the first convention is
the lesser evil. So the GAP library follows the book, and if you call
the SpaceGroup
function you will get the same space group
representatives as given there. This does not cause any problems as long
as you do calculations within these groups treating them just as matrix
groups of certain isomorphism types. However, if it is necesary to
consider the action of a space group as affine mappings on the natural
lattice, you need to use the transposed representation of the space
group. For this purpose the library offers the TransposedSpaceGroup
function which not only transposes the matrices, but also adapts
appropriately the associated group presentation.
Both these functions are described in detail in the following.
The library functions
NrCrystalFamilies( dim )
NrCrystalFamilies
returns the number of crystal families in case of
dimension dim. It can be used to formulate loops over the crystal
families.
There are 4, 6, and 23 crystal families of dimension 2, 3, and 4, respectively.
gap> n := NrCrystalFamilies( 4 ); 23
DisplayCrystalFamily( dim, family )
DisplayCrystalFamily
displays for the specified crystal family
essentially the same information as is provided for that family in Table
1 of BBNWZ78, namely
item the family name,
vspace-2mm
item the number of parameters,
vspace-2mm
item the common rational decomposition pattern,
vspace-2mm
item the common real decomposition pattern,
vspace-2mm
item the number of crystal systems in the family, and
vspace-2mm
item the number of Bravais flocks in the family.
vspace-2mm
For details see BBNWZ78.
gap> DisplayCrystalFamily( 4, 17 ); #I Family XVII: cubic orthogonal; 2 free parameters; #I Q-decomposition pattern 1+3; R-decomposition pattern 1+3; #I 2 crystal systems; 6 Bravais flocks gap> DisplayCrystalFamily( 4, 18 ); #I Family XVIII: octagonal; 2 free parameters; #I Q-irreducible; R-decomposition pattern 2+2; #I 1 crystal system; 1 Bravais flock gap> DisplayCrystalFamily( 4, 21 ); #I Family XXI: di-isohexagonal orthogonal; 1 free parameter; #I R-irreducible; 2 crystal systems; 2 Bravais flocks
NrCrystalSystems( dim )
NrCrystalSystems
returns the number of crystal systems in case of
dimension dim. It can be used to formulate loops over the crystal
systems.
There are 4, 7, and 33 crystal systems of dimension 2, 3, and 4, respectively.
gap> n := NrCrystalSystems( 2 ); 4
The following two functions are functions of crystal systems.
Each crystal system is characterized by a pair (dim,,system)
where
dim is the associated dimension, and system is the number of the
crystal system.
DisplayCrystalSystem( dim, system )
DisplayCrystalSystem
displays for the specified crystal system
essentially the same information as is provided for that system in Table
1 of BBNWZ78, namely
item the number of Q-classes in the crystal system and
vspace-2mm
item the identification number, i.,e., the tripel
(dim,,system,,q-class)
described below, of the Q-class
that is the holohedry of the crystal system.
vspace-2mm
For details see BBNWZ78.
gap> for sys in [ 1 .. 4 ] do DisplayCrystalSystem( 2, sys ); od; #I Crystal system 1: 2 Q-classes; holohedry (2,1,2) #I Crystal system 2: 2 Q-classes; holohedry (2,2,2) #I Crystal system 3: 2 Q-classes; holohedry (2,3,2) #I Crystal system 4: 4 Q-classes; holohedry (2,4,4)
NrQClassesCrystalSystem( dim, system )
NrQClassesCrystalSystem
returns the number of Q-classes within the
given crystal system. It can be used to formulate loops over the
Q-classes.
The following five functions are functions of Q-classes.
In general, the parameters characterizing a Q-class will form a
triple (dim,,system,,q-class)
where dim is the associated
dimension, system is the number of the associated crystal system, and
q-class is the number of the Q-class within the crystal system.
However, in case of dimensions 2 or 3, a Q-class may also be
characterized by a pair (dim, IT-number)
where IT-number is the
number in the International Tables for Crystallography Hah83 of
any space-group type lying in (a Z-class of) that Q-class, or
just by the Hermann-Mauguin symbol of any space-group type lying in (a
Z-class of) that Q-class.
The Hermann-Mauguin symbols indexHermann-Mauguin symbol which we use
in GAP are the short Hermann-Mauguin symbols defined in the 1983
edition of the International Tables Hah83, but any occurring
indices are expressed by ordinary integers, and bars are replaced by
minus signs. For example, the Hermann-Mauguin symbol
Poverline{4}2_1m will be represented by the string "P-421m"
.
DisplayQClass( dim, system, q-class )
DisplayQClass( dim, IT-number )
DisplayQClass( Hermann-Mauguin-symbol )
DisplayQClass
displays for the specified Q-class essentially the
same information as is provided for that Q-class in Table 1 of
BBNWZ78 (except for the defining relations given there), namely
item the size of the groups in the Q-class,
vspace-2mm
item the isomorphism type of the groups in the Q-class,
vspace-2mm
item the Hurley pattern,
vspace-2mm
item the rational constituents,
vspace-2mm
item the number of Z-classes in the Q-class, and
vspace-2mm
item the number of space-group types in the Q-class.
vspace-2mm
For details see BBNWZ78.
gap> DisplayQClass( "p2" ); #I Q-class H (2,1,2): size 2; isomorphism type 2.1 = C2; #I Q-constituents 2*(2,1,2); cc; 1 Z-class; 1 space group gap> DisplayQClass( "R-3" ); #I Q-class (3,5,2): size 6; isomorphism type 6.1 = C6; #I Q-constituents (3,1,2)+(3,4,3); ncc; 2 Z-classes; 2 space grps gap> DisplayQClass( 3, 195 ); #I Q-class (3,7,1): size 12; isomorphism type 12.5 = A4; #I C-irreducible; 3 Z-classes; 5 space grps gap> DisplayQClass( 4, 27, 4 ); #I Q-class H (4,27,4): size 20; isomorphism type 20.3 = D10xC2; #I Q-irreducible; 1 Z-class; 1 space group gap> DisplayQClass( 4, 29, 1 ); #I *Q-class (4,29,1): size 18; isomorphism type 18.3 = D6xC3; #I R-irreducible; 3 Z-classes; 5 space grps
Note in the preceding examples that, as pointed out above, the term ``size'' denotes the order of a representative group of the specified Q-class and, of course, not the (infinite) class length.
FpGroupQClass( dim, system, q-class )
FpGroupQClass( dim, IT-number )
FpGroupQClass( Hermann-Mauguin-symbol )
FpGroupQClass
returns a finitely presented group F, say, which is
isomorphic to the groups in the specified Q-class.
The presentation of that group is the same as the corresponding
presentation given in Table 1 of BBNWZ78 except for the fact that
its generators are listed in reverse order. The reason for this change
is that, whenever the group in question is solvable, the resulting
generators form an AG system (as defined in GAP) if they are numbered
``from the top to the bottom'', and the presentation is a polycyclic
power commutator presentation. The AgGroupQClass
function described
next will make use of this fact in order to construct an ag group
isomorphic to F.
Note that, for any Z-class in the specified Q-class, the matrix
group returned by the MatGroupZClass
function (see below) not only is
isomorphic to F, but also its generators satisfy the defining relators
of F.
Besides of the usual components, the group record of F will have an
additional component F.crQClass
which saves a list of the parameters
that specify the given Q-class.
gap> F := FpGroupQClass( 4, 20, 3 ); FpGroupQClass( 4, 20, 3 ) gap> F.generators; [ f.1, f.2 ] gap> F.relators; [ f.1^2*f.2^-3, f.2^6, f.2^-1*f.1^-1*f.2*f.1*f.2^-4 ] gap> F.size; 12 gap> F.crQClass; [ 4, 20, 3 ]
AgGroupQClass( dim, system, q-class )
AgGroupQClass( dim, IT-number )
AgGroupQClass( Hermann-Mauguin-symbol )
AgGroupQClass
returns an ag group A, say, isomorphic to the groups in
the specified Q-class, if these groups are solvable, or the value
false
(together with an appropriate warning), otherwise.
A is constructed by first establishing a finitely presented group (as
it would be returned by the FpGroupQClass
function described above) and
then constructing from it an isomorphic ag group. If the underlying
AG system is not yet a PAG system (see sections More about Ag Words and
More about Ag Groups), it will be refined appropriately (and a warning
will be displayed).
Besides of the usual components, the group record of A will have an
additional component A.crQClass
which saves a list of the parameters
that specify the given Q-class.
gap> A := AgGroupQClass( 4, 31, 3 ); #I Warning: a non-solvable group can't be represented as an ag group false gap> A := AgGroupQClass( 4, 20, 3 ); #I Warning: the presentation has been extended to get a PAG system AgGroupQClass( 4, 20, 3 ) gap> A.generators; [ f.1, f.21, f.22 ] gap> A.size; 12 gap> A.crQClass; [ 4, 20, 3 ]
CharTableQClass( dim, system, q-class )
CharTableQClass( dim, IT-number )
CharTableQClass( Hermann-Mauguin-symbol )
CharTableQClass
returns the character table T, say, of a
representative group of (a Z-class of) the specified Q-class.
Although the set of characters can be considered as an invariant of the specified Q-class, the resulting table will depend on the order in which GAP sorts the conjugacy classes of elements and the irreducible characters and hence, in general, will not coincide with the corresponding table presented in BBNWZ78.
CharTableQClass
proceeds as follows. If the groups in the given
Q-class are solvable, then it first calls the AgGroupQClass
and
RefinedAgSeries
functions to get an isomorphic ag group with a
PAG system, and then it calls the CharTable
function to compute the
character table of that ag group. In the case of one of the five
Q-classes of dimension 4 whose groups are not solvable, it first calls
the FpGroupQClass
function to get an isomorphic finitely presented
group, then it constructs a specially chosen faithful permutation
representation of low degree for that group, and finally it determines
the character table of the resulting permutation group again by calling
the CharTable
function.
In general, the above strategy will be much more efficient than the
alternative possibilities of calling the CharTable
function for a
finitely presented group provided by the FpGroupQClass
function or for
a matrix group provided by the MatGroupZClass
function.
gap> T := CharTableQClass( 4, 20, 3 );; gap> DisplayCharTable( T ); CharTableQClass( 4, 20, 3 )2 2 1 1 2 2 2 3 1 1 1 1 . .
1a 3a 6a 2a 4a 4b 2P 1a 3a 3a 1a 2a 2a 3P 1a 1a 2a 2a 4b 4a 5P 1a 3a 6a 2a 4a 4b
X.1 1 1 1 1 1 1 X.2 1 1 1 1 -1 -1 X.3 1 1 -1 -1 A -A X.4 1 1 -1 -1 -A A X.5 2 -1 1 -2 . . X.6 2 -1 -1 2 . .
A = E(4) = ER(-1) = i
NrZClassesQClass( dim, system, q-class )
NrZClassesQClass( dim, IT-number )
NrZClassesQClass( Hermann-Mauguin-symbol )
NrZClassesQClass
returns the number of Z-classes within the given
Q-class. It can be used to formulate loops over the Z-classes.
The following functions are functions of Z-classes.
In general, the parameters characterizing a Z-class will form a
quadruple (dim,,system, mboxq-class,,z-class)
where dim
is the associated dimension, system is the number of the associated
crystal system, q-class is the number of the associated Q-class
within the crystal system, and z-class is the number of the Z-class
within the Q-class. However, in case of dimensions 2 or 3, a
Z-class may also be characterized by a pair (dim, IT-number)
where IT-number is the number in the International Tables Hah83
of any space-group type lying in that Z-class, or just by the
Hermann-Mauguin symbol of any space-group type lying in that Z-class.
DisplayZClass( dim, system, q-class, z-class )
DisplayZClass( dim, IT-number )
DisplayZClass( Hermann-Mauguin-symbol )
DisplayZClass
displays for the specified Z-class essentially the
same information as is provided for that Z-class in Table 1 of
BBNWZ78 (except for the generating matrices of a class
representative group given there), namely
item for dimensions 2 and 3, the Hermann-Mauguin symbol of a
representative space-group type which belongs to that Z-class,
vspace-2mm
item the Bravais type,
vspace-2mm
item some decomposability information,
vspace-2mm
item the number of space-group types belonging to the Z-class,
vspace-2mm
item the size of the associated cohomology group.
vspace-2mm
For details see BBNWZ78.
gap> DisplayZClass( 2, 3 ); #I Z-class (2,2,1,1) = Z(pm): Bravais type II/I; fully Z-reducible; #I 2 space groups; cohomology group size 2 gap> DisplayZClass( "F-43m" ); #I Z-class (3,7,4,2) = Z(F-43m): Bravais type VI/II; Z-irreducible; #I 2 space groups; cohomology group size 2 gap> DisplayZClass( 4, 2, 3, 2 ); #I Z-class B (4,2,3,2): Bravais type II/II; Z-decomposable; #I 2 space groups; cohomology group size 4 gap> DisplayZClass( 4, 21, 3, 1 ); #I *Z-class (4,21,3,1): Bravais type XVI/I; Z-reducible; #I 1 space group; cohomology group size 1
MatGroupZClass( dim, system, q-class, z-class )
MatGroupZClass( dim, IT-number )
MatGroupZClass( Hermann-Mauguin-symbol )
MatGroupZClass
returns a dim times dim matrix group M, say, which
is a representative of the specified Z-class. Its generators satisfy
the defining relators of the finitely presented group which may be
computed by calling the FpGroupQClass
function (see above) for the
Q-class which contains the given Z-class.
The generators of M are the same matrices as those given in Table 1 of
BBNWZ78. Note, however, that they will be listed in reverse order
to keep them in parallel to the abstract generators provided by the
FpGroupQClass
function (see above).
Besides of the usual components, the group record of M will have an
additional component M.crZClass
which saves a list of the parameters
that specify the given Z-class. (In fact, in order to make the
resulting group record consistent with those returned by the
NormalizerZClass
or ZClassRepsDadeGroup
functions described below, it
also will have an additional component M.crConjugator
containing just
the identity element of M.)
gap> M := MatGroupZClass( 4, 20, 3, 1 ); MatGroupZClass( 4, 20, 3, 1 ) gap> for g in M.generators do > Print( "\n" ); PrintArray( g ); od; Print( "\n" );[ [ 0, 1, 0, 0 ], [ -1, 0, 0, 0 ], [ 0, 0, -1, -1 ], [ 0, 0, 0, 1 ] ]
[ [ -1, 0, 0, 0 ], [ 0, -1, 0, 0 ], [ 0, 0, -1, -1 ], [ 0, 0, 1, 0 ] ]
gap> M.size; 12 gap> M.crZClass; [ 4, 20, 3, 1 ]
NormalizerZClass( dim, system, q-class, z-class )
NormalizerZClass( dim, IT-number )
NormalizerZClass( Hermann-Mauguin-symbol )
NormalizerZClass
returns the normalizer N, say, in GL(dim,Z) of
the representative dim times dim matrix group which is constructed by
the MatGroupZClass
function (see above).
If the size of N is finite, then N again lies in some Z-class. In
this case, the group record of N will contain two additional components
N.crZClass
and N.crConjugator
which provide the parameters of
that Z-class and a matrix g in GL(dim,Z), respectively, such
that N = g^{-1} R g, where R is the representative group of that
Z-class.
gap> N := NormalizerZClass( 4, 20, 3, 1 ); NormalizerZClass( 4, 20, 3, 1 ) gap> for g in N.generators do > Print( "\n" ); PrintArray( g ); od; Print( "\n" );[ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, -1, -1 ] ]
[ [ 1, 0, 0, 0 ], [ 0, -1, 0, 0 ], [ 0, 0, -1, -1 ], [ 0, 0, 1, 0 ] ]
[ [ 0, 1, 0, 0 ], [ -1, 0, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ]
[ [ -1, 0, 0, 0 ], [ 0, -1, 0, 0 ], [ 0, 0, -1, 0 ], [ 0, 0, 0, -1 ] ]
gap> N.size; 96 gap> N.crZClass; [ 4, 20, 22, 1 ] gap> N.crConjugator = N.identity; true
gap> L := NormalizerZClass( 3, 42 ); NormalizerZClass( 3, 3, 2, 4 ) gap> L.size; 16 gap> L.crZClass; [ 3, 4, 7, 2 ] gap> L.crConjugator; [ [ 0, 0, -1 ], [ 1, 0, 0 ], [ 0, -1, -1 ] ] gap> M := NormalizerZClass( "C2/m" ); Group( [ [ -1, 0, 0 ], [ 0, -1, 0 ], [ 0, 0, -1 ] ], [ [ 0, -1, 0 ], [ -1, 0, 0 ], [ 0, 0, -1 ] ], [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 1 ] ], [ [ -1, 0, 0 ], [ 0, -1, 0 ], [ -1, -1, 1 ] ], [ [ 0, 1, -1 ], [ 1, 0, -1 ], [ 0, 0, -1 ] ] ) gap> M.size; "infinity" gap> IsBound( M.crZClass ); false
NrSpaceGroupTypesZClass( dim, system, q-class, z-class )
NrSpaceGroupTypesZClass( dim, IT-number )
NrSpaceGroupTypesZClass( Hermann-Mauguin-symbol )
NrSpaceGroupTypes
returns the number of space-group types within the
given Z-class. It can be used to formulate loops over the space-group
types.
gap> N := NrSpaceGroupTypesZClass( 4, 4, 1, 1 ); 13
Some of the Z-classes of dimension d, say, are ``maximal'' in the sense that the groups in these classes are maximal finite subgroups of GL(d,Z). Generalizing a term which is being used for dimension 4, we call the representatives of these maximal Z-classes the ``Dade groups'' of dimension d.
NrDadeGroups( dim )
NrDadeGroups
returns the number of Dade groups of dimension dim. It
can be used to formulate loops over the Dade groups.
There are 2, 4, and 9 Dade groups of dimension 2, 3, and 4, respectively.
gap> NrDadeGroups( 4 ); 9
DadeGroup( dim, n )
DadeGroup
returns the nth Dade group of dimension dim.
gap> D := DadeGroup( 4, 7 ); MatGroupZClass( 4, 31, 7, 2 )
DadeGroupNumbersZClass( dim, system, q-class, z-class )
DadeGroupNumbersZClass( dim, IT-number )
DadeGroupNumbersZClass( Hermann-Mauguin-symbol )
DadeGroupNumbersZClass
returns the set of all those integers n_i for
which the n_ith Dade group of dimension dim contains a subgroup
which, in GL(dim,Z), is conjugate to the representative group of the
given Z-class.
gap> dadeNums := DadeGroupNumbersZClass( 4, 4, 1, 2 ); [ 1, 5, 8 ] gap> for d in dadeNums do > D := DadeGroup( 4, d ); > Print( D, " of size ", Size( D ), "\n" ); > od; MatGroupZClass( 4, 20, 22, 1 ) of size 96 MatGroupZClass( 4, 30, 13, 1 ) of size 288 MatGroupZClass( 4, 32, 21, 1 ) of size 384
ZClassRepsDadeGroup( dim, system, q-class, z-class, n )
ZClassRepsDadeGroup( dim, IT-number, n )
ZClassRepsDadeGroup( Hermann-Mauguin-symbol, n )
ZClassRepsDadeGroup
determines in the nth Dade group of dimension
dim all those conjugacy classes whose groups are, in GL(dim,Z),
conjugate to the Z-class representative group R, say, of the given
Z-class. It returns a list of representative groups of these
conjugacy classes.
Let M be any group in the resulting list. Then the group record of M
provides two components M.crZClass
and M.crConjugator
which
contain the list of Z-class parameters of R and a suitable matrix
g from GL(dim,Z), respectively, such that M equals g^{-1} R g.
gap> DadeGroupNumbersZClass( 2, 2, 1, 2 ); [ 1, 2 ] gap> ZClassRepsDadeGroup( 2, 2, 1, 2, 1 ); [ MatGroupZClass( 2, 2, 1, 2 )^[ [ 0, 1 ], [ -1, 0 ] ] ] gap> ZClassRepsDadeGroup( 2, 2, 1, 2, 2 ); [ MatGroupZClass( 2, 2, 1, 2 )^[ [ 1, -1 ], [ 0, -1 ] ], MatGroupZClass( 2, 2, 1, 2 )^[ [ 1, 0 ], [ -1, 1 ] ] ] gap> R := last[2];; gap> R.crZClass; [ 2, 2, 1, 2 ] gap> R.crConjugator; [ [ 1, 0 ], [ -1, 1 ] ]
The following functions are functions of space-group types.
In general, the parameters characterizing a space-group type will form a
quintuple (dim, system,,q-class,,z-class,,sg-type)
where
dim is the associated dimension, system is the number of the
associated crystal system, q-class is the number of the associated
Q-class within the crystal system, z-class is the number of the
Z-class within the Q-class, and sg-type is the space-group type
within the Z-class. However, in case of dimensions 2 or 3, you may
instead specify a Z-class by a pair (dim, IT-number)
or by its
Hermann-Mauguin symbol (as described above). Then the function will
handle the first space-group type within that Z-class, i.,e.,
sg-type = 1, that is, the corresponding symmorphic space group (split
extension).
DisplaySpaceGroupType( dim, system, q-class, z-class, sg-type
)
DisplaySpaceGroupType( dim, IT-number )
DisplaySpaceGroupType( Hermann-Mauguin-symbol )
DisplaySpaceGroupType
displays for the specified space-group type some
of the information which is provided for that space-group type in Table 1
of BBNWZ78, namely
item the orbit size associated with that space-group type and,
vspace-2mm
item for dimensions 2 and 3, the IT-number and the Hermann-Mauguin
symbol.
vspace-2mm
For details see BBNWZ78.
gap> DisplaySpaceGroupType( 2, 17 ); #I Space-group type (2,4,4,1,1); IT(17) = p6mm; orbit size 1 gap> DisplaySpaceGroupType( "Pm-3" ); #I Space-group type (3,7,2,1,1); IT(200) = Pm-3; orbit size 1 gap> DisplaySpaceGroupType( 4, 32, 10, 2, 4 ); #I *Space-group type (4,32,10,2,4); orbit size 18 gap> DisplaySpaceGroupType( 3, 6, 1, 1, 4 ); #I *Space-group type (3,6,1,1,4); IT(169) = P61, IT(170) = P65; #I orbit size 2; fp-free
DisplaySpaceGroupGenerators( dim, system, q-class, z-class,
sg-type )
DisplaySpaceGroupGenerators( dim, IT-number )
DisplaySpaceGroupGenerators( Hermann-Mauguin-symbol )
DisplaySpaceGroupGenerators
displays the non-translation generators of
a representative space group of the specified space-group type without
actually constructing that matrix group.
In more details: Let n = dim be the given dimension, and let M_1,
ldots, M_r be the generators of the representative n times n matrix
group of the given Z-class (this is the group which you will get if
you call the MatGroupZClass
function (see above) for that Z-class).
Then, for the given space-group type, the SpaceGroup
function described
below will construct as representative of that space-group type an (n+1)
times (n+1) matrix group which is generated by the n translations
which are induced by the (standard) basis vectors of the n-dimensional
Euclidian space, and r additional matrices S_1, ldots, S_r of the
form S_i = left[ catcode`|=12 begin{tabular}{c|c} catcode`|=13
M_i & t_i \ hline 0 & 1 end{tabular} right], where the n times
n submatrices M_i are as defined above, and the t_i are n-columns
with rational entries. The DisplaySpaceGroupGenerators
function saves
time by not constructing the group, but just displaying the r matrices
S_1,
ldots, S_r.
gap> DisplaySpaceGroupGenerators( "P61" ); #I The non-translation generators of SpaceGroup( 3, 6, 1, 1, 4 ) are[ [ -1, 0, 0, 0 ], [ 0, -1, 0, 0 ], [ 0, 0, 1, 1/2 ], [ 0, 0, 0, 1 ] ]
[ [ 0, -1, 0, 0 ], [ 1, -1, 0, 0 ], [ 0, 0, 1, 1/3 ], [ 0, 0, 0, 1 ] ]
SpaceGroup( dim, system, q-class, z-class, sg-type )
SpaceGroup( dim, IT-number )
SpaceGroup( Hermann-Mauguin-symbol )
SpaceGroup
returns a (dim+1) times (dim+1 ) matrix group S, say,
which is a representative of the given space-group type (see also the
description of the DisplaySpaceGroupGenerators
function above).
gap> S := SpaceGroup( "P61" ); SpaceGroup( 3, 6, 1, 1, 4 ) gap> for s in S.generators do > Print( "\n" ); PrintArray( s ); od; Print( "\n" );[ [ -1, 0, 0, 0 ], [ 0, -1, 0, 0 ], [ 0, 0, 1, 1/2 ], [ 0, 0, 0, 1 ] ]
[ [ 0, -1, 0, 0 ], [ 1, -1, 0, 0 ], [ 0, 0, 1, 1/3 ], [ 0, 0, 0, 1 ] ]
[ [ 1, 0, 0, 1 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ]
[ [ 1, 0, 0, 0 ], [ 0, 1, 0, 1 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ]
[ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 1 ], [ 0, 0, 0, 1 ] ]
gap> S.crSpaceGroupType; [ 3, 6, 1, 1, 4 ]
Besides of the usual components, the resulting group record of S
contains an additional component S.crSpaceGroupType
which saves a
list of the parameters that specify the given space-group type.
Moreover, it contains, in form of a finitely presented group, a presentation of S which is satisfied by the matrix generators. If the factor group of S by its translation normal subgroup is solvable then this presentation is chosen such that it is a polycyclic power commutator presentation. The proper way to access this presentation is to call the following function.
FpGroup( S )
FpGroup
returns a finitely presented group G, say, which is
isomorphic to S, where S is expected to be a space group. It is
chosen such that there is an isomrphism from G to S which maps each
generator of G onto the corresponding generator of S. This means, in
particular, that the matrix generators of S satisfy the relators of
G.
gap> G := FpGroup( S ); Group( g1, g2, g3, g4, g5 ) gap> for rel in G.relators do Print( rel, "\n" ); od; g1^2*g5^-1 g2^3*g5^-1 g2^-1*g1^-1*g2*g1 g3^-1*g1^-1*g3*g1*g3^2 g3^-1*g2^-1*g3*g2*g4*g3^2 g4^-1*g1^-1*g4*g1*g4^2 g4^-1*g2^-1*g4*g2*g4*g3^-1 g4^-1*g3^-1*g4*g3 g5^-1*g1^-1*g5*g1 g5^-1*g2^-1*g5*g2 g5^-1*g3^-1*g5*g3 g5^-1*g4^-1*g5*g4 gap> # Verify that the matrix generators of S satisfy the relators of G. gap> ForAll( G.relators, > rel -> MappedWord( rel, G.generators, S.generators ) = S.identity ); true
TransposedSpaceGroup( dim, system, q-class, z-class, sg-type )
TransposedSpaceGroup( dim, IT-number )
TransposedSpaceGroup( Hermann-Mauguin-symbol )
TransposedSpaceGroup( S )
TransposedSpaceGroup
returns a matrix group T, say, whose generators
are just the transposed generators (in the same order) of the
corresponding space group S specified by the arguments. As for S, you
may get a finite presentation for T via the FpGroup
function.
The purpose of this function is explicitly discussed in the introduction to this section.
gap> T := TransposedSpaceGroup( S ); TransposedSpaceGroup( 3, 6, 1, 1, 4 ) gap> for m in T.generators do > Print( "\n" ); PrintArray( m ); od; Print( "\n" ); [ [ -1, 0, 0, 0 ], [ 0, -1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 1/2, 1 ] ] [ [ 0, 1, 0, 0 ], [ -1, -1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 1/3, 1 ] ] [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 1, 0, 0, 1 ] ] [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 1, 0, 1 ] ] [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 1, 1 ] ]
GAP 3.4.4