73.130 InducedCat1

InducedCat1( C, iota )

When {cal C} is the induced cat1-group associated to {cal X} the induced cat1-group may be obtained by construction the induced crossed module and then using the Cat1XMod function. An experimental, alternative procedure is to calculate the induced cat1-group iota_ast G = G ast_R Q directly. This has been implemented for the case when {cal C} = ( e;t,h : G to R) and iota : R to Q is an inclusion.

The output from the calculation is a cat1-group {cal C}_ast = (e_ast;t_ast, h_ast : iota_{ast}G to Q) together with a morphism of crossed modules {cal C} to {cal C}_{ast}.

In the example an induced cat1-group is constructed whose associated crossed module has source c4 times c4 and range d16, so the source of the cat1-group is d16 semidirect (c4 times c4).

    gap> CDX := Cat1XMod( DX );
    cat1-group [Perm(d8 

X c4) ==> d8] 
    gap> inc := InclusionMorphism( d8, d16 );;
    gap> ICDX := InducedCat1( CDX, inc );
          

    new perm group size 256
    cat1-group <ICG([Perm(d8 

X c4) ==> d8])> 
    gap> XICDX := XModCat1( ICDX );
    Crossed module [ker(<ICG([Perm(d8 

X c4) ==> d8])>)->d16]
    gap> AbelianInvariants( XICDX.source );
    [ 4, 4 ]  

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