73.128 InducedXMod

InducedXMod( X, iota )

InducedXMod( Q, P, M )

This function requires as data a conjugation crossed module {cal X} = (partial : M to P) and a homomorphism iota : P to Q. This data may be specified using either of the two forms shown, where the latter form required Q ge P ge M.

In the first example, iota is a surjection from d8 to k4.

    gap> d8gen := d8.generators;
    [ (1,3,5,7)(2,4,6,8), (1,3)(4,8)(5,7) ]
    gap> k4gen := k4.generators;
    [ (1,2)(3,4), (1,3)(2,4) ]
    gap> DX;
    Crossed module [c4->d8]
    gap> iota := GroupHomomorphismByImages( d8, k4, d8gen, k4gen );;
    gap> IDXsurj := InducedXMod( DX, iota );
    Crossed module [c4/ker->k4
    gap> XModPrint( IDXsurj );
    Crossed module [c4/ker->k4] :- 
    : Source group c4/ker has generators:
      [ (1,2,3,4) ]
    : Range group has parent ( s4 ) and has generators:
      [ (1,2)(3,4), (1,3)(2,4) ]
    : Boundary homomorphism maps source generators to:
      [ ( 1, 2)( 3, 4) ]
    : Action homomorphism maps range generators to automorphisms:
      (1,2)(3,4) --> { source gens --> [ (1,2,3,4) ] }
      (1,3)(2,4) --> { source gens --> [ (1,4,3,2) ] }
      These 2 automorphisms generate the group of automorphisms.
    : Induced XMod from Crossed module [c4->d8]  with source morphism:
      [ (1,3,5,7)(2,4,6,8) ]
       --> [ (1,2,3,4) ]    

In a second example we take (c3 -> s3) as the initial crossed module and s3 -> s4 as the inclusion. The induced group turns out to be the special linear group sl(2,3).

    gap> s3 := Subgroup( s4, [ (2,3), (1,2,3) ] );;
    gap> c3 := Subgroup( s3, [ (1,2,3) ] );
    gap> s3.name := "s3";;  c3.name := "c3";;
    gap> InducedXMod( s4, s3, c3 );

    Action of RQ on generators of I :- 
      (1,2,3,4) : (1,7,6,3)(2,8,5,4)
      (1,2) : (1,2)(3,4)(5,8)(6,7)
    

I  Protecting the first 1 generators.
    

I  there are 2 generators and 3 relators of total length 12
    Simplified presentation for I :- 
    

I  generators: [ fI.1, fI.5 ]
    

I  relators:
    

I  1.  3  [ 2, 2, 2 ]
    

I  2.  3  [ 1, 1, 1 ]
    

I  3.  6  [ 2, -1, -2, 1, -2, -1 ]

     I has Size: 24
    **************** 
     Searching Solvable Groups Library:
     GroupId = 
    rec(
      catalogue := [ 24, 14 ],
      names := [ "SL(2,3)" ],
      size := 24 )
    Image of I has index 2 in RQ and is generated by : 
    [ (1,2,3), (1,2,4), (1,4,3), (2,3,4) ]

    Crossed module [i*(c3)->s4]            

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