Before describing general functions for computing induced structures, we consider coproducts of crossed modules which provide induced crossed modules in certain cases.
Need to add here a reference (or two) for coproducts.
‣ CoproductXMod( X1, X2 ) | ( operation ) |
This function calculates the coproduct crossed module of crossed modules \({\mathcal X}_1 = (\partial_1 : S_1 \to R)\) and \({\mathcal X}_2 = (\partial_2 : S_2 \to R)\) which have a common range \(R\). The source \(S_2\) of \({\mathcal X}_2\) acts on \(S_1\) via \(\partial_2\) and the action of \({\mathcal X}_1\), so we can form a precrossed module \((\partial' : S_1 \ltimes S_2 \to R)\) where \(\partial'(s_1,s_2) = (\partial_1 s_1)(\partial_2 s_2)\). The action of this precrossed module is the diagonal action \((s_1,s_2)^r = (s_1^r,s_2^r)\). Factoring out by the Peiffer subgroup, we obtain the coproduct crossed module \({\mathcal X}_1 \circ {\mathcal X}_2\).
In the example the structure descriptions of the precrossed module, the Peiffer subgroup, and the resulting coproduct are printed out when InfoLevel(InfoXMod} is at least \(1\). The coproduct comes supplied with attribute CoproductInfo, which includes the embedding morphisms of the two factors.
gap> q8 := Group( (1,2,3,4)(5,8,7,6), (1,5,3,7)(2,6,4,8) );; gap> X8 := XModByAutomorphismGroup( q8 );; gap> s4 := Range( X8 );; gap> SetName( q8, "q8" ); SetName( s4, "s4" ); gap> k4 := NormalSubgroups( s4 )[3];; SetName( k4, "k4" ); gap> Z8 := XModByNormalSubgroup( s4, k4 );; gap> SetName( X8, "X8" ); SetName( Z8, "Z8" ); gap> SetInfoLevel( InfoXMod, 1 ); gap> XZ8 := CoproductXMod( X8, Z8 ); #I prexmod is [ "C2 x C2 x Q8", "S4" ] #I peiffer subgroup is C2 #I the coproduct is [ "C2 x C2 x C2 x C2", "S4" ] [Group( [ f1, f2, f3, f4 ] )->s4] gap> SetName( XZ8, "XZ8" ); gap> info := CoproductInfo( XZ8 ); rec( embeddings := [ [X8 => XZ8], [Z8 => XZ8] ], xmods := [ X8, Z8 ] )
‣ InducedXMod( args ) | ( function ) |
‣ InducedCat1( args ) | ( function ) |
‣ IsInducedXMod( xmod ) | ( property ) |
‣ MorphismOfInducedXMod( xmod ) | ( attribute ) |
A morphism of crossed modules \((\sigma, \rho) : {\mathcal X}_1 \to {\mathcal X}_2\) factors uniquely through an induced crossed module \(\rho_{\ast} {\mathcal X}_1 = (\delta : \rho_{\ast} S_1 \to R_2)\). Similarly, a morphism of cat1-groups factors through an induced cat1-group. Calculation of induced crossed modules of \({\mathcal X}\) also provides an algebraic means of determining the homotopy \(2\)-type of homotopy pushouts of the classifying space of \({\mathcal X}\). For more background from algebraic topology see references in [BH78], [BW95], [BW96]. Induced crossed modules and induced cat1-groups also provide the building blocks for constructing pushouts in the categories XMod and Cat1.
Data for the cases of algebraic interest is provided by a conjugation crossed module \({\mathcal X} = (\partial : S \to R)\) and a homomorphism \(\iota\) from \(R\) to a third group \(Q\). (It is hoped to implement more general cases in due course.) The output from the calculation is a crossed module \(\iota_{\ast}{\mathcal X} = (\delta : \iota_{\ast}S \to Q)\) together with a morphism of crossed modules \({\mathcal X} \to \iota_{\ast}{\mathcal X}\). When \(\iota\) is a surjection with kernel \(K\) then \(\iota_{\ast}S = [S,K]\) (see [BH78]). When \(\iota\) is an inclusion the induced crossed module may be calculated using a copower construction [BW95] or, in the case when \(R\) is normal in \(Q\), as a coproduct of crossed modules ([BW96], but not yet implemented). When \(\iota\) is neither a surjection nor an inclusion, \(\iota\) is factored 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. These constructions use Tietze transformation routines in the library file tietze.gi.
As a first, surjective example, we take for \({\mathcal X}\) the normal inclusion crossed module of a4 in s4, and for \(\iota\) the surjection from s4 to s3 with kernel k4. The induced crossed module is isomorphic to X3.
gap> s4gens := GeneratorsOfGroup( s4 ); [ (1,2), (2,3), (3,4) ] gap> a4gens := GeneratorsOfGroup( a4 ); [ (1,2,3), (2,3,4) ] gap> s3b := Group( (5,6),(6,7) );; SetName( s3b, "s3b" ); gap> epi := GroupHomomorphismByImages( s4, s3b, s4gens, [(5,6),(6,7),(5,6)] );; gap> X4 := XModByNormalSubgroup( s4, a4 );; gap> indX4 := SurjectiveInducedXMod( X4, epi ); [a4/ker->s3b] gap> Display( indX4 ); Crossed module [a4/ker->s3b] :- : Source group a4/ker has generators: [ (1,3,2), (1,2,3) ] : Range group s3b has generators: [ (5,6), (6,7) ] : Boundary homomorphism maps source generators to: [ (5,6,7), (5,7,6) ] : Action homomorphism maps range generators to automorphisms: (5,6) --> { source gens --> [ (1,2,3), (1,3,2) ] } (6,7) --> { source gens --> [ (1,2,3), (1,3,2) ] } These 2 automorphisms generate the group of automorphisms. gap> morX4 := MorphismOfInducedXMod( indX4 ); [[a4->s4] => [a4/ker->s3b]]
For a second, injective example we take for \({\mathcal X}\) a conjugation crossed module.
gap> d8 := Subgroup( d16, [ b1^2, b2 ] ); SetName( d8, "d8" ); Group([ (11,13,15,17)(12,14,16,18), (12,18)(13,17)(14,16) ]) gap> c4 := Subgroup( d8, [ b1^2 ] ); SetName( c4, "c4" ); Group([ (11,13,15,17)(12,14,16,18) ]) gap> Y16 := XModByNormalSubgroup( d16, d8 ); [d8->d16] gap> Y8 := SubXMod( Y16, c4, d8 ); [c4->d8] gap> inc8 := InclusionMorphism2DimensionalDomains( Y16, Y8 ); [[c4->d8] => [d8->d16]] gap> incd8 := RangeHom( inc8 );; gap> indY8 := InducedXMod( Y8, incd8 ); #I induced group has Size: 16 #I factor 2 is abelian with invariants: [ 4, 4 ] i*([c4->d8]) gap> morY8 := MorphismOfInducedXMod( indY8 ); [[c4->d8] => i*([c4->d8])]
For a third example we take the identity mapping on s3c as boundary, and the inclusion of s3c in s4 as \(\iota\). The induced group is a general linear group GL(2,3).
gap> s3c := Subgroup( s4, [ (2,3), (3,4) ] );; gap> SetName( s3c, "s3c" ); gap> indXs3c := InducedXMod( s4, s3c, s3c ); #I induced group has Size: 48 i*([s3c->s3c]) gap> StructureDescription( indXs3c ); [ "GL(2,3)", "S4" ]
‣ AllInducedXMods( Q ) | ( operation ) |
This function calculates all the induced crossed modules InducedXMod( Q, P, M ), where P runs over all conjugacy classes of subgroups of Q and M runs over all non-trivial subgroups of P.
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