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10 Utility functions
 10.1 Printing Lists
 10.2 Inclusion and Restriction Mappings
 10.3 Abelian Modules
 10.4 Distinct and Common Representatives

10 Utility functions

By a utility function we mean a GAP function which is

10.1 Printing Lists

10.1-1 PrintListOneItemPerLine
‣ PrintListOneItemPerLine( L )( operation )

This function is used in Chapter 4 when printing image lists of derivations and sections.


gap> L := [ [1,2,3,4], true, [ (1,2), (2,3) ] ];; 
gap> PrintListOneItemPerLine( L ); 
[ [ 1, 2, 3, 4 ], 
  true, 
  [ (1,2), (2,3) ] 
  ]

10.2 Inclusion and Restriction Mappings

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

10.2-1 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) ] -> [ () ]

10.2-2 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

10.3 Abelian Modules

10.3-1 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> 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.


10.4 Distinct and Common Representatives

10.4-1 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) ]

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