A morphism of crossed modules (sigma, rho) : {cal X}_1 to {cal X}_2 factors uniquely through an induced crossed module rho_{ast} {cal X}_1 = (delta : rho_{ast} S_1 to R_2). Similarly, a morphism of cat1-groups factors through an induced cat1-group. Calculation of induced crossed modules of {cal X} also provides an algebraic means of determining the homotopy 2-type of homotopy pushouts of the classifying space of {cal X}. For more background from algebraic topology see references in xmodBH1, xmodBW1, xmodBW2. Induced crossed modules and induced cat1-groups also provide the building blocks for constructing pushouts in the categories textbfXMod and textbfCat1.
Data for the cases of algebraic interest is provided by a conjugation crossed module {cal X} = (partial : S to R) and a homomorphism iota from R to a third group Q. The output from the calculation is a crossed module iota_{ast}{cal X} = (delta : iota_{ast}S to Q) together with a morphism of crossed modules {cal X} to iota_{ast}{cal X}. When iota is a surjection with kernel K then iota_{ast} S = [S,K] (see xmodBH1). When iota is an inclusion the induced crossed module may be calculated using a copower construction xmodBW1 or, in the case when R is normal in Q, as a coproduct of crossed modules (xmodBW2, not yet implemented). When iota is neither a surjection nor an inclusion, iota is written as the composite of the surjection onto the image and the inclusion of the image in Q, and then the composite induced crossed module is constructed.
Other functions required by the induced crossed module construction
include a function to produce a common transversal for the left and
right cosets of a subgroup (see refIsCommonTransversal and
refCommonTransversal). Also, modifications to some of the Tietze
transformation routines in fptietze.g are required. These have yet
to be released as part of the GAP library and so are made available
in this package in file felsch.g, but are not documented here.
As a simple example we take for {cal X} the conjugation crossed module (partial : c4 to d8) and for iota the inclusion of d8 in d16. The induced crossed module has c4 times c4 as source.
gap> d16 := DihedralGroup( 16 ); d16.name := "d16";;
Group( (1,2,3,4,5,6,7,8), (2,8)(3,7)(4,6) )
gap> d8 := Subgroup( d16, [ (1,3,5,7)(2,4,6,8), (1,3)(4,8)(5,7) ] );;
gap> c4 := Subgroup( d8, [ (1,3,5,7)(2,4,6,8) ] );;
gap> d8.name := "d8";; c4.name := "c4";;
gap> DX := ConjugationXMod( d8, c4 );
Crossed module [c4->d8]
gap> iota := InclusionMorphism( d8, d16 );;
gap> IDXincl := InducedXMod( DX, iota );
Action of RQ on generators of I :-
(1,2,3,4,5,6,7,8) : (1,4)(2,3)
(2,8)(3,7)(4,6) : (1,2)(3,4)
#I Protecting the first 1 generators.
#I there are 2 generators and 3 relators of total length 12
partitioning the generators: [ [ 2 ], [ 1 ] ]
Simplified presentation for I :-
#I generators: [ fI.1, fI.3 ]
#I relators:
#I 1. 4 [ 1, 1, 1, 1 ]
#I 2. 4 [ 2, 2, 2, 2 ]
#I 3. 4 [ 2, -1, -2, 1 ]I has Size: 16 ****************
Group is abelian factor 1 is abelian with invariants: [ 4 ] factor 2 is abelian with invariants: [ 4 ] Image of I has index 4 in RQ and is generated by : [ ( 1, 3, 5, 7)( 2, 4, 6, 8), ( 1, 7, 5, 3)( 2, 8, 6, 4) ]
gap> XModPrint( IDXincl ); Crossed module [i*(c4)->d16] :- : Source group i*(c4) has generators: [ ( 1, 2, 4, 7)( 3, 5, 8,11)( 6, 9,12,14)(10,13,15,16), ( 1, 3, 6,10)( 2, 5, 9,13)( 4, 8,12,15)( 7,11,14,16) ] : Range group = d16 has generators: [ (1,2,3,4,5,6,7,8), (2,8)(3,7)(4,6) ] : Boundary homomorphism maps source generators to: [ ( 1, 3, 5, 7)( 2, 4, 6, 8), ( 1, 7, 5, 3)( 2, 8, 6, 4) ] : Action homomorphism maps range generators to automorphisms: (1,2,3,4,5,6,7,8) --> { source gens --> [ ( 1,10, 6, 3)( 2,13, 9, 5)( 4,15,12, 8)( 7,16,14,11), ( 1, 7, 4, 2)( 3,11, 8, 5)( 6,14,12, 9)(10,16,15,13) ] } (2,8)(3,7)(4,6) --> { source gens --> [ ( 1, 7, 4, 2)( 3,11, 8, 5)( 6,14,12, 9)(10,16,15,13), ( 1,10, 6, 3)( 2,13, 9, 5)( 4,15,12, 8)( 7,16,14,11) ] } These 2 automorphisms generate the group of automorphisms. : Kernel of the crossed module has generators: [ ( 1, 5,12,16)( 2, 8,14,10)( 3, 9,15, 7)( 4,11, 6,13) ] : Induced XMod from Crossed module [c4->d8] with source morphism: [ (1,3,5,7)(2,4,6,8) ] --> [ ( 1, 2, 4, 7)( 3, 5, 8,11)( 6, 9,12,14)(10,13,15,16) ]
In some of the sections which follow the output is very lengthy and so
has been pruned.
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GAP 3.4.4