73.117 AutomorphismPermGroup for crossed modules

XModOps.AutomorphismPermGroup( X )

This function constructs a permutation group PermAut(X) isomorphic to the group of automorphisms of the crossed module X. First the automorphism groups of the source and range of X are obtained and AutomorphismPair used to obtain permutation representations of these. The direct product of these permutation groups is constructed, and the required automorphism group is a subgroup of this direct product. The result is stored as X.automorphismPermGroup which has fields defining the various embeddings and projections.

    gap> autXSC := AutomorphismPermGroup( XSC );
    PermAut([c3->s3])
    gap> autXSC.projsrc;
    GroupHomomorphismByImages( PermAut([c3->s3]), PermAut(c3), 
    [ (5,6,7), (1,2)(3,4)(6,7) ], [ (), (1,2) ] )
    gap> autXSC.projrng;
    GroupHomomorphismByImages( PermAut([c3->s3]), PermAut(s3), 
    [ (5,6,7), (1,2)(3,4)(6,7) ], [ (3,4,5), (1,2)(4,5) ] )
    gap> autXSC.embedSourceAuto;
    GroupHomomorphismByImages( PermAut(c3), PermAut(c3)xPermAut(s3),
        [ (1,2) ], [ (1,2) ] )
    gap> autXSC.embedRangeAuto;
    GroupHomomorphismByImages( PermAut(s3), PermAut(c3)xPermAut(s3), 
    [ (3,5,4), (1,2)(4,5) ], [ (5,7,6), (3,4)(6,7) ] )
    gap> autXSC.autogens;
    [ [ GroupHomomorphismByImages( c3, c3, [ (1,2,3)(4,6,5) ],
            [ (1,2,3)(4,6,5) ] ), GroupHomomorphismByImages( s3, s3, 
            [ (4,5,6), (2,3)(5,6) ], [ (4,5,6), (2,3)(4,5) ] ) ], 
      [ GroupHomomorphismByImages( c3, c3, [ (1,2,3)(4,6,5) ], 
            [ (1,3,2)(4,5,6) ] ), GroupHomomorphismByImages( s3, s3, 
            [ (4,5,6), (2,3)(5,6) ], [ (4,6,5), (2,3)(5,6) ] ) ] ]   

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