chap10.txt

  
  10 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.
  
  
  10.1 Inclusion and Restriction Mappings
  
  These functions have been moved to the gpd package, but are still documented
  here.
  
  10.1-1 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.
  
    Example  
    
    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) ] -> [ () ]
    
  
  
  10.1-2 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.
  
    Example  
    
    gap> autd8 := AutomorphismGroup( d8 );;
    gap> innd8 := InnerAutomorphismsByNormalSubgroup( d8, d8 );;
    gap> GeneratorsOfGroup( innd8 );
    [ ^(1,2,3,4), ^(1,3) ]
    gap> IsGroupOfAutomorphisms( innd8 );
    true
    
  
  
  
  10.2 Abelian Modules
  
  10.2-1 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.
  
  The    resulting    Xabmod    is    isomorphic    to    the    output   from
  XModByAutomorphismGroup( k4 );.
  
    Example  
    
    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.