Quelle grpsupp.tex Sprache: Latech

%
\Chapter{Supportive 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?'

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Predefined groups}

All groups of order $2$ to $32$ are predefined. They can be accessed
by variables of the kind `GTW<o>\_{<n>}' where defines the order
of the group and <n> the number of the group of order <o> as they
appear in \cite{thomaswood80:GT} . For example, `GTW16\_6' defines the
group of Thomas -- Wood type $16/6$, which is actually $D_4 \times
C_2$.

Alternatively, these groups can be accessed via the function

\>TWGroup( <o>, <n> )

with <o> and <n> as above. In addition, all these groups are stored in the
list {\tt GroupList}.

Conversely, for any group <G> of order at most 32,

\>IdTWGroup( <G> )

returns a pair `[<o>,<n>]', meaning that is isomorphic to the group
$o/n$.

\beginexample
 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 ]
\endexample

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Operation tables for groups}

\>PrintTable( <G> )

`PrintTable' prints the Cayley table of the group .

\beginexample
 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

\endexample

Sometimes different symbols for the elements in the would make the table
look nicer. For the group $4/2$ ($\Z_2 \times \Z_2$) one could choose
the canonical form as pairs of zeros and ones.

\beginexample
 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)
\endexample

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Group endomorphisms}

\>Endomorphisms( <G> )

`Endomorphisms' computes all the endomorphisms of the group .
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.

\beginexample
 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) ] ]
\endexample

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Group automorphisms}

\>Automorphisms( <G> )

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

\beginexample
 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) ] ]
\endexample

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Inner automorphisms of a group}

\>InnerAutomorphisms( <G> )

`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.

\beginexample
 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) ]
\endexample

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Isomorphic groups}

\>IsIsomorphicGroup( <G>, <H> )

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

\beginexample
 gap> IsIsomorphicGroup( SymmetricGroup( 4 ), GTW24_12 );
 true
\endexample

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Subgroups of a group}

\>Subgroups( <G> )

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

\beginexample
 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) ]) ]
\endexample

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Normal subgroups generated by a single element}

\>OneGeneratedNormalSubgroups( <G> )

`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'.

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Invariant subgroups}

\>IsInvariantUnderMaps( <G>, , <maps> )

For a list of mappings, <maps> on the group <G> and
a subgroup of <G>, `IsInvariantUnderMaps' returns the truth
value of `` 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.

\beginexample
 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 ]) ]
\endexample

\>IsCharacteristicSubgroup( <G>, )

A subgroup of the group <G> is *characteristic* if it is invariant under
all automorphisms on <G>. For a subgroup of the group <G>,
`IsCharacteristicSubgroup' returns the truth value of `` is a characteristic
subgroup of <G>''. If the group is defined as the subgroup of a group
<G> then the function call

\>IsCharacteristicInParent( )

has the same result.

\beginexample
 gap> IsCharacteristicInParent( Centre( GTW16_11 ) );
 true
\endexample

\>IsFullinvariant( <G>, )

A subgroup of the group <G> is *fully invariant* if it is invariant under
all endomorphisms on <G>.
For a subgroup of the group <G>, `IsFullinvariant' returns the
truth value of `` is a fully invariant subgroup of <G>''.

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

If the group is defined as the subgroup of a group <G> then the function
call

\>IsFullinvariantInParent( )

has the same result.

\beginexample
 gap> IsFullinvariantInParent( Centre( GTW16_11 ) );
 true
\endexample

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Coset representatives}

\>RepresentativesModNormalSubgroup( <G>, <N> )

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>.

\beginexample
 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 ]
\endexample

\>NontrivialRepresentativesModNormalSubgroup( <G>, <N> )

This function behaves as `RepresentativesModNormalSubgroup' but it excludes
the representative for the congruence class which contains the neutral element
of the group.

\beginexample
 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 ]
\endexample

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Scott length}

\>ScottLength( <G> )

The function `ScottLength' retuns the Scott-length of the group .
For a definition of the Scott-length of a group and an idea for an
algorithm for the general case see \cite{scott69:TAOPMOAGATSOCPPGI}.
In the case of a class 2 nilpotent finite group <G> a faster algorithm
described in \cite{ecker98:OTNOPFONGOC2} is used.

\beginexample
 gap> ScottLength( GTW6_2 );
 2
 gap> ScottLength( GTW16_11 );
 4
\endexample

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Other useful functions for groups}

\>AsPermGroup( <G> )

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.

\beginexample
 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) ])
\endexample

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