- Predefined groups
- Operation tables for groups
- Group endomorphisms
- Group automorphisms
- Inner automorphisms of a group
- Isomorphic groups
- Subgroups of a group
- Normal subgroups generated by a single element
- Invariant subgroups
- Coset representatives
- Scott length
- Other useful functions for groups

In order to support nearring calculations, a few functions for groups had to be added to the standard GAP group library functions.

The functions described here can be found in the source files
`grpend.g?`

and `grpsupp.g?`

All groups of order 2 to 32 are predefined. They can be accessed
by variables of the kind `GTW`

`o``_`

`n`` where `

`o` defines the order
of the group and `n` the number of the group of order `o` as they
appear in <[>thomaswood80:GT] . For example, `GTW16_6`

defines the
group of Thomas -- Wood type 16/6, which is actually *D*_{4} ×*C*_{2}.

Alternatively, these groups can be accessed via the function

`TWGroup( `

`, `

` )`

with `o` and `n` as above. In addition, all these groups are stored in the
list ttGroupList.

Conversely, for any group `G` of order at most 32,

`IdTWGroup( `

` )`

returns a pair `[`

`o``,`

`n``]`

, meaning that `G` is isomorphic to the group
*o*/*n*.

gap> G := GTW6_2; 6/2 gap> H := TWGroup( 4, 2 ); 4/2 gap> D := DirectProduct( G, H ); Group([ (1,2), (1,2,3), (4,5), (6,7) ]) gap> IdTWGroup( D ); [ 24, 4 ] gap> GroupList; [ 2/1, 3/1, 4/1, 4/2, 5/1, 6/1, 6/2, 7/1, 8/1, 8/2, 8/3, 8/4, 8/5, 9/1, 9/2, 10/1, 10/2, 11/1, 12/1, 12/2, 12/3, 12/4, 12/5, 13/1, 14/1, 14/2, 15/1, 16/1, 16/2, 16/3, 16/4, 16/5, 16/6, 16/7, 16/8, 16/9, 16/10, 16/11, 16/12, 16/13, 16/14, 17/1, 18/1, 18/2, 18/3, 18/4, 18/5, 19/1, 20/1, 20/2, 20/3, 20/4, 20/5, 21/1, 21/2, 22/1, 22/2, 23/1, 24/1, 24/2, 24/3, 24/4, 24/5, 24/6, 24/7, 24/8, 24/9, 24/10, 24/11, 24/12, 24/13, 24/14, 24/15, 25/1, 25/2, 26/1, 26/2, 27/1, 27/2, 27/3, 27/4, 27/5, 28/1, 28/2, 28/3, 28/4, 29/1, 30/1, 30/2, 30/3, 30/4, 31/1, 32/1, 32/2, 32/3, 32/4, 32/5, 32/6, 32/7, 32/8, 32/9, 32/10, 32/11, 32/12, 32/13, 32/14, 32/15, 32/16, 32/17, 32/18, 32/19, 32/20, 32/21, 32/22, 32/23, 32/24, 32/25, 32/26, 32/27, 32/28, 32/29, 32/30, 32/31, 32/32, 32/33, 32/34, 32/35, 32/36, 32/37, 32/38, 32/39, 32/40, 32/41, 32/42, 32/43, 32/44, 32/45, 32/46, 32/47, 32/48, 32/49, 32/50, 32/51 ]

`PrintTable( `

` )`

`PrintTable`

prints the Cayley table of the group `G`.

gap> G := GTW4_2; 4/2 gap> PrintTable( G ); Let: g0 := () g1 := (3,4) g2 := (1,2) g3 := (1,2)(3,4) * | g0 g1 g2 g3 ------------------ g0 | g0 g1 g2 g3 g1 | g1 g0 g3 g2 g2 | g2 g3 g0 g1 g3 | g3 g2 g1 g0

Sometimes different symbols for the elements in the would make the table
look nicer. For the group 4/2 (**Z**_{2} ×**Z**_{2}) one could choose
the canonical form as pairs of zeros and ones.

gap> G := GTW4_2; 4/2 gap> SetSymbols( G, ["(0,0)","(1,0)","(0,1)","(1,1)"] ); gap> PrintTable( G ); Let: (0,0) := () (1,0) := (3,4) (0,1) := (1,2) (1,1) := (1,2)(3,4) * | (0,0) (1,0) (0,1) (1,1) ----------------------------------- (0,0) | (0,0) (1,0) (0,1) (1,1) (1,0) | (1,0) (0,0) (1,1) (0,1) (0,1) | (0,1) (1,1) (0,0) (1,0) (1,1) | (1,1) (0,1) (1,0) (0,0)

`Endomorphisms( `

` )`

`Endomorphisms`

computes all the endomorphisms of the group `G`.
This function is most essential for computing the nearrings on a group.
The endomorphisms are returned as a list of group homomorphisms. So all
functions for mappings and homomorphisms are applicable.

gap> G := TWGroup( 4, 2 ); 4/2 gap> Endomorphisms( G ); [ [ (1,2), (3,4) ] -> [ (), () ], [ (1,2), (3,4) ] -> [ (), (1,2) ], [ (1,2), (3,4) ] -> [ (), (3,4) ], [ (1,2), (3,4) ] -> [ (), (1,2)(3,4) ], [ (1,2), (3,4) ] -> [ (1,2), () ], [ (1,2), (3,4) ] -> [ (3,4), () ], [ (1,2), (3,4) ] -> [ (1,2)(3,4), () ], [ (1,2), (3,4) ] -> [ (1,2), (1,2) ] , [ (1,2), (3,4) ] -> [ (3,4), (3,4) ], [ (1,2), (3,4) ] -> [ (1,2)(3,4), (1,2)(3,4) ], [ (1,2), (3,4) ] -> [ (1,2), (3,4) ], [ (1,2), (3,4) ] -> [ (1,2)(3,4), (3,4) ], [ (1,2), (3,4) ] -> [ (3,4), (1,2) ], [ (1,2), (3,4) ] -> [ (1,2)(3,4), (1,2) ], [ (1,2), (3,4) ] -> [ (3,4), (1,2)(3,4) ], [ (1,2), (3,4) ] -> [ (1,2), (1,2)(3,4) ] ]

`Automorphisms( `

` )`

`Automorphisms`

computes all the automorphisms of the group `G`.
The automorphisms are returned as a list of group homomorphisms. So all
functions for mappings and homomorphisms are applicable.

gap> Automorphisms( GTW4_2 ); [ IdentityMapping( 4/2 ), [ (1,2), (3,4) ] -> [ (1,2)(3,4), (3,4) ], [ (1,2), (3,4) ] -> [ (3,4), (1,2) ], [ (3,4), (1,2) ] -> [ (1,2), (1,2)(3,4) ], [ (3,4), (1,2) ] -> [ (1,2)(3,4), (3,4) ], [ (3,4), (1,2) ] -> [ (1,2)(3,4), (1,2) ] ]

`InnerAutomorphisms( `

` )`

`InnerAutomorphisms`

computes all the inner automorphisms of the group
`G`.
The inner automorphisms are returned as a list of group homomorphisms. So all
functions for mappings and homomorphisms are applicable.

gap> InnerAutomorphisms( AlternatingGroup( 4 ) ); [ ^(), ^(2,3,4), ^(2,4,3), ^(1,2)(3,4), ^(1,2,3), ^(1,2,4), ^(1,3,2), ^(1,3,4), ^(1,3)(2,4), ^(1,4,2), ^(1,4,3), ^(1,4)(2,3) ]

`IsIsomorphicGroup( `

`, `

` )`

`IsIsomorphicGroup`

determines if the groups `G` and `H` are
isomorphic. If they are isomorphic, an isomorphism between these two groups
can be found with `IsomorphismGroups`

.

gap> IsIsomorphicGroup( SymmetricGroup( 4 ), GTW24_12 ); true

`Subgroups( `

` )`

`Subgroups`

returns a list of all subgroups of the group `G`, if there
are only finitely many subgroups.

gap> Subgroups( TWGroup( 8, 4 ) ); [ Group(()), Group([ (1,3)(2,4) ]), Group([ (2,4) ]), Group([ (1,3) ]), Group([ (1,2)(3,4) ]), Group([ (1,4)(2,3) ]), Group([ (1,3)(2,4), (2,4) ]), Group([ (1,3)(2,4), (1,2,3,4) ]), Group([ (1,3)(2,4), (1,2)(3,4) ]), Group([ (1,3)(2,4), (2,4), (1,2,3,4) ]) ]

`OneGeneratedNormalSubgroups( `

` )`

`OneGeneratedSubgroups`

returns a list of all proper, non-trivial normal
subgroups of the group `G` which are generated by one element.
`OneGeneratedSubgroups`

is a synonym for `GeneratorsOfCongruenceLattice`

.

gap> OneGeneratedNormalSubgroups( AlternatingGroup(4) ); [ Group([ (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) ]) ]

`IsInvariantUnderMaps( `

`, `

`, `

` )`

For a list of mappings, `maps` on the group `G` and
a subgroup `U` of `G`, `IsInvariantUnderMaps`

returns the truth
value of ```U` is invariant under all mappings in `maps`''. In the following
example this function is used to compute all fully invariant subgroups of
the dihedral group of order 12.

gap> D12 := DihedralGroup( 12 ); <pc group of size 12 with 3 generators> gap> s := Subgroups( D12 ); [ Group([ ]), Group([ f1 ]), Group([ f1*f3^2 ]), Group([ f1*f3 ]), Group([ f2*f3 ]), Group([ f1*f2 ]), Group([ f1*f2*f3^2 ]), Group([ f1*f2*f3 ]), Group([ f3 ]), Group([ f1, f2*f3 ]), Group([ f1*f3^2, f2*f3 ]), Group([ f1*f3, f2*f3 ]), Group([ f3, f1 ]), Group([ f3, f2 ]), Group([ f3, f1*f2 ]), Group([ f3, f1, f2 ]) ] gap> e := Endomorphisms( D12 );; gap> f := Filtered( s, sg -> IsInvariantUnderMaps( D12, sg, e ) ); [ Group([ ]), Group([ f3 ]), Group([ f3, f1, f2 ]) ]

`IsCharacteristicSubgroup( `

`, `

` )`

A subgroup `U` of the group `G` is **characteristic** if it is invariant under
all automorphisms on `G`. For a subgroup `U` of the group `G`,
`IsCharacteristicSubgroup`

returns the truth value of ```U` is a characteristic
subgroup of `G`''. If the group `U` is defined as the subgroup of a group
`G` then the function call

`IsCharacteristicInParent( `

` )`

has the same result.

gap> IsCharacteristicInParent( Centre( GTW16_11 ) ); true

`IsFullinvariant( `

`, `

` )`

A subgroup `U` of the group `G` is **fully invariant** if it is invariant under
all endomorphisms on `G`.
For a subgroup `U` of the group `G`, `IsFullinvariant`

returns the
truth value of ```U` is a fully invariant subgroup of `G`''.

gap> G := GTW6_2; 6/2 gap> S := Subgroup( G, [(1,2)] ); Group([ (1,2) ]) gap> IsFullinvariant( G, S ); false

If the group `U` is defined as the subgroup of a group `G` then the function
call

`IsFullinvariantInParent( `

` )`

has the same result.

gap> IsFullinvariantInParent( Centre( GTW16_11 ) ); true

`RepresentativesModNormalSubgroup( `

`, `

` )`

If `G` is a group and `N` is a normal subgroup of `G` then the function
`RepresentativesModNormalSubgroup`

returns a set of representatives for
the congruence classes modulo the normal subgroup `N`, i.e. a set of elements
of `G` with exactly one element from each cogruence class modulo `N`.

gap> G := DihedralGroup( 16 ); <pc group of size 16 with 4 generators> gap> C := Centre( G ); Group([ f4 ]) gap> RepresentativesModNormalSubgroup( G, C ); [ <identity> of ..., f1, f2, f3, f2*f3, f1*f2*f4, f1*f3*f4, f1*f2*f3*f4 ]

`NontrivialRepresentativesModNormalSubgroup( `

`, `

` )`

This function behaves as `RepresentativesModNormalSubgroup`

but it excludes
the representative for the congruence class which contains the neutral element
of the group.

gap> G := DihedralGroup( 16 ); <pc group of size 16 with 4 generators> gap> C := Centre( G ); Group([ f4 ]) gap> NontrivialRepresentativesModNormalSubgroup( G, C ); [ f1, f2, f3, f2*f3, f1*f2*f4, f1*f3*f4, f1*f2*f3*f4 ]

`ScottLength( `

` )`

The function `ScottLength`

retuns the Scott-length of the group `G`.
For a definition of the Scott-length of a group and an idea for an
algorithm for the general case see <[>scott69:TAOPMOAGATSOCPPGI].
In the case of a class 2 nilpotent finite group `G` a faster algorithm
described in <[>ecker98:OTNOPFONGOC2] is used.

gap> ScottLength( GTW6_2 ); 2 gap> ScottLength( GTW16_11 ); 4

`AsPermGroup( `

` )`

For a group `G`, `AsPermGroup`

returns a permutation group that is isomorphic
to `G`. In the case of a permutation group this is the group itself.

gap> D24 := DihedralGroup( 24 ); <pc group of size 24 with 4 generators> gap> D24p := AsPermGroup( D24 ); <permutation group of size 24 with 4 generators> gap> IsomorphismGroups( D24, D24p ); [ f1, f2, f3, f4 ] -> [ (1,17)(2,16)(3,18)(4,14)(5,13)(6,15)(7,20)(8,19)(9,21)(10,22)(11,24)(12,23), (1,11,4,9,2,12,5,7,3,10,6,8)(13,23,16,21,14,24,17,19,15,22,18,20), (1,4,2,5,3,6)(7,10,8,11,9,12)(13,16,14,17,15,18)(19,22,20,23,21,24), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24) ] gap> C12 := CyclicGroup( 12 ); <pc group of size 12 with 3 generators> gap> AsPermGroup( C12 ); Group([ ( 1, 7, 4,10, 2, 8, 5,11, 3, 9, 6,12), ( 1, 4, 2, 5, 3, 6)( 7,10, 8,11, 9,12), ( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12) ])

SONATA manual

October 2018