GAP (XMod) - Chapter 8: Utility functions

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### 8. Utility functions

By a utility function we mean a {\GAP} function which is

- needed by other functions in this package,

- not (as far as we know) provided by the standard GAP library,

- more suitable for inclusion in the main library than in this package.

#### 8.1 Inclusion and Restriction Mappings

##### 8.1-1 InclusionMappingGroups

`> InclusionMappingGroups`( G, H ) | ( operation )

`> RestrictionMappingGroups`( hom, src, rng ) | ( operation )

`> MappingToOne`( G, H ) | ( operation )

The first set of utilities concerns inclusion and restriction mappings. Restriction may apply to both the source and the range of the map. The map `incd8` is the inclusion of `d8` in `d16` used in Section 3.4.

    gap> Print( incd8, "\n" );
    [ (11,13,15,17)(12,14,16,18), (11,18)(12,17)(13,16)(14,15) ] ->
    [ (11,13,15,17)(12,14,16,18), (11,18)(12,17)(13,16)(14,15) ]
    gap> imd8 := Image( incd8 );;
    gap> resd8 := RestrictionMappingGroups( incd8, c4, imd8 );;
    gap> Source( res8 );  Range( res8 );
    c4
    Group([ (11,13,15,17)(12,14,16,18), (11,18)(12,17)(13,16)(14,15) ])
    gap> MappingToOne( c4, imd8 );
    [ (11,13,15,17)(12,14,16,18) ] -> [ () ]

#### 8.2 Endomorphism Classes and Automorphisms

##### 8.2-1 EndomorphismClasses

`> EndomorphismClasses`( grp, case ) | ( function )

`> EndoClassNaturalHom`( class ) | ( attribute )

`> EndoClassIsomorphism`( class ) | ( attribute )

`> EndoClassConjugators`( class ) | ( attribute )

`> AutoGroup`( class ) | ( attribute )

The monoid of endomorphisms of a group is used when calculating the monoid of derivations of a crossed module and when determining all the cat1-structures on a group.

An endomorphism epsilon of R with image H' is determined by

- a normal subgroup N of R and a permutation representation theta : R/N -> Q of the quotient, giving a projection theta circ nu : R -> Q, where nu : R -> R/N is the natural homomorphism;

- an automorphism alpha of Q;

- a subgroup H' in a conjugacy class [H] of subgroups of R isomorphic to Q having representative H, an isomorphism phi : Q cong H, and a conjugating element c in R such that H^c = H'.

Then epsilon takes values

\[ \epsilon r ~=~ (\phi\alpha\theta\nu\,r)^c~. \]

Endomorphisms are placed in the same class if they have the same choice of N and [H], and so the number of endomorphisms is

\[ |{\rm End}(R)| ~=~ \sum_{{\rm classes}} |{\rm Aut}(Q)|.|[H]|~. \]

The function `EndomorphismClasses( <grp>, <case> )` may be called in three ways:

- case 1 includes automorphisms and the zero map,

- case 2 excludes automorphisms and the zero map,

- case 3 is when `N` intersects `H` trivially.

    gap> end8 := EndomorphismClasses( d8, 1 );;
    gap> Length( end8 );
    13
    gap> e4 := end8[4];
    <enumerator>
    gap> EndoClassNaturalHom( e4 );
    GroupHomomorphismByImages( d8, Group( [ f1 ] ),
    [ (11,13,15,17)(12,14,16,18), (12,18)(13,17)(14,16) ], [ f1, f1 ] )
    gap> EndoClassIsomorphism( e4 );
    Pcgs([ f1 ]) -> [ (11,13)(14,18)(15,17) ]
    gap> EndoClassConjugators( e4 );
    [ (), (12,18)(13,17)(14,16) ]
    gap> AutoGroup( e4 );
    Group( [ Pcgs([ f1 ]) -> [ f1 ] ] )
    gap> L := List( end8, e -> Length(EndoClassConjugators(e)) * Size(AutoGroup(e)) );
    [ 8, 1, 2, 2, 1, 2, 2, 1, 2, 2, 6, 6, 1 ]
    gap> Sum( L );
    36

##### 8.2-2 InnerAutomorphismByNormalSubgroup

`> InnerAutomorphismByNormalSubgroup`( G, N ) | ( operation )

`> IsGroupOfAutomorphisms`( A ) | ( property )

Inner automorphisms of a group `G` by the elements of a normal subgroup `N` are calculated with the first of these functions, usually with `G` = `N`.

    gap> autd8 := AutomorphismGroup( d8 );;
    gap> innd8 := InnerAutomorphismsByNormalSubgroup( d8, d8 );;
    gap> GeneratorsOfGroup( innd8 );
    [ InnerAutomorphism( d8, (11,13,15,17)(12,14,16,18) ),
      InnerAutomorphism( d8, (12,18)(13,17)(14,16) ) ]
    gap> IsGroupOfAutomorphisms( innd8 );
    true

#### 8.3 Abelian Modules

##### 8.3-1 AbelianModuleObject

`> AbelianModuleObject`( grp, act ) | ( operation )

`> IsAbelianModule`( obj ) | ( property )

`> AbelianModuleGroup`( obj ) | ( attribute )

`> AbelianModuleAction`( obj ) | ( attribute )

An abelian module is an abelian group together with a group action. These are used by the crossed module constructor `XModByAbelianModule`.

The resulting `Xabmod` is isomorphic to the output from `XModByAutomorphismGroup( k4 );`.

    gap> x := (6,7)(8,9);;  y := (6,8)(7,9);;  z := (6,9)(7,8);;
    gap> k4 := Group( x, y );  SetName( k4, "k4" );
    gap> s3 := Group( (1,2), (2,3) );;  SetName( s3, "s3" );
    gap> alpha := GroupHomomorphismByImages( k4, k4, [x,y], [y,x] );
    gap> beta := GroupHomomorphismByImages( k4, k4, [x,y], [x,z] );
    gap> aut := Group( alpha, beta );
    gap> act := GroupHomomorphismByImages( s3, aut, [(1,2),(2,3)], [alpha,beta] );
    gap> abmod := AbelianModuleObject( k4, act );
    &lt;enumerator&rt;
    gap> Xabmod := XModByAbelianModule( abmod );
    [k4->s3]

#### 8.4 Distinct and Common Representatives

##### 8.4-1 DistinctRepresentatives

`> DistinctRepresentatives`( list ) | ( operation )

`> CommonRepresentatives`( list ) | ( operation )

`> CommonTransversal`( grp, subgrp ) | ( operation )

`> IsCommonTransversal`( grp, subgrp, list ) | ( operation )

The final set of utilities deal with lists of subsets of [1 ... n] and construct systems of distinct and common representatives using simple, non-recursive, combinatorial algorithms.

When L is a set of n subsets of [1 ... n] and the Hall condition is satisfied (the union of any k subsets has at least k elements), a set of distinct representatives exists.

When J,K are both lists of n sets, the function `CommonRepresentatives` returns two lists: the set of representatives, and a permutation of the subsets of the second list. It may also be used to provide a common transversal for sets of left and right cosets of a subgroup H of a group G, although a greedy algorithm is usually quicker.

    gap> J := [ [1,2,3], [3,4], [3,4], [1,2,4] ];;
    gap> DistinctRepresentatives( J );
    [ 1, 3, 4, 2 ]
    gap> K := [ [3,4], [1,2], [2,3], [2,3,4] ];;
    gap> CommonRepresentatives( J, K );
    [ [ 3, 3, 3, 1 ], [ 1, 3, 4, 2 ] ]
    gap> CommonTransversal( d16, c4 );
    [ (), (12,18)(13,17)(14,16), (11,12,13,14,15,16,17,18),
      (11,12)(13,18)(14,17)(15,16) ]
    gap> IsCommonTransversal( d16, c4, [ (), c, d, c*d ] );
    true

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