GAP (XMod) - Chapter 11: Utility functions

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11 Utility functions
11.1 Inclusion and Restriction Mappings
11.1-1 InclusionMappingGroups
11.1-2 InnerAutomorphismsByNormalSubgroup
11.2 Abelian Modules
11.2-1 AbelianModuleObject

### 11 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.

Sections on Printing Lists and Distinct and Common Representatives were moved to the Utils package with version 2.56.

#### 11.1 Inclusion and Restriction Mappings

These functions have been moved to the gpd package, but are still documented here.

##### 11.1-1 InclusionMappingGroups

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

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

This set of utilities concerns mappings. The map `incd8` is the inclusion of `d8` in `d16` used in Section 3.4. `MappingToOne(G,H)` maps the whole of \(G\) to the identity element in \(H\).

    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> MappingToOne( c4, imd8 );
    [ (11,13,15,17)(12,14,16,18) ] -> [ () ]

##### 11.1-2 InnerAutomorphismsByNormalSubgroup

`‣ InnerAutomorphismsByNormalSubgroup`( 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 );
    [ ^(1,2,3,4), ^(1,3) ]
    gap> IsGroupOfAutomorphisms( innd8 );
    true

#### 11.2 Abelian Modules

##### 11.2-1 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` (2.1-1).

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> k4a := Group( x, y );;  SetName( k4a, "k4a" );
    gap> gens3a := [ (1,2), (2,3) ];;
    gap> s3a := Group( gens3a );;  SetName( s3a, "s3a" );
    gap> alpha := GroupHomomorphismByImages( k4a, k4a, [x,y], [y,x] );;
    gap> beta := GroupHomomorphismByImages( k4a, k4a, [x,y], [x,z] );;
    gap> auta := Group( alpha, beta );;
    gap> acta := GroupHomomorphismByImages( s3a, auta, gens3a, [alpha,beta] );;
    gap> abmod := AbelianModuleObject( k4a, acta );;
    gap> Xabmod := XModByAbelianModule( abmod );
    [k4a->s3a]
    gap> Display( Xabmod );

    Crossed module [k4a->s3a] :-
    : Source group k4a has generators:
      [ (6,7)(8,9), (6,8)(7,9) ]
    : Range group s3a has generators:
      [ (1,2), (2,3) ]
    : Boundary homomorphism maps source generators to:
      [ (), () ]
    : Action homomorphism maps range generators to automorphisms:
      (1,2) --> { source gens --> [ (6,8)(7,9), (6,7)(8,9) ] }
      (2,3) --> { source gens --> [ (6,7)(8,9), (6,9)(7,8) ] }
      These 2 automorphisms generate the group of automorphisms.

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