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 ) |
‣ CoproductInfo( X0 ) | ( attribute ) |
This function calculates the coproduct crossed module of two or more crossed modules which have a common range \(R\). The standard method applies to \(\calX_1 = (\partial_1 : S_1 \to R)\) and \(\calX_2 = (\partial_2 : S_2 \to R)\). See below for the case of three or more crossed modules.
The source \(S_2\) of \(\calX_2\) acts on \(S_1\) via \(\partial_2\) and the action of \(\calX_1\), 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 \(\calX_1 \circ \calX_2\).
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> XAq8 := XModByAutomorphismGroup( q8 );; gap> s4b := Range( XAq8 );; gap> SetName( q8, "q8" ); SetName( s4b, "s4b" ); gap> a := q8.1;; b := q8.2;; gap> alpha := GroupHomomorphismByImages( q8, q8, [a,b], [a^-1,b] );; gap> beta := GroupHomomorphismByImages( q8, q8, [a,b], [a,b^-1] );; gap> k4b := Subgroup( s4b, [ alpha, beta ] );; SetName( k4b, "k4b" ); gap> Z8 := XModByNormalSubgroup( s4b, k4b );; gap> SetName( XAq8, "XAq8" ); SetName( Z8, "Z8" ); gap> SetInfoLevel( InfoXMod, 1 ); gap> XZ8 := CoproductXMod( XAq8, Z8 ); #I prexmod is [ [ 32, 47 ], [ 24, 12 ] ] #I peiffer subgroup is C2, [ 2, 1 ] #I the coproduct is [ "C2 x C2 x C2 x C2", "S4" ], [ [ 16, 14 ], [ 24, 12 ] ] [Group( [ f1, f2, f3, f4 ] )->s4b] gap> SetName( XZ8, "XZ8" ); gap> info := CoproductInfo( XZ8 ); rec( embeddings := [ [XAq8 => XZ8], [Z8 => XZ8] ], xmods := [ XAq8, Z8 ] ) gap> SetInfoLevel( InfoXMod, 0 );
Given a list of more than two crossed modules with a common range \(R\), then an iterated coproduct is formed:
\[ \bigcirc~\left[ \calX_1,\calX_2,\ldots,\calX_n\right] ~=~ \calX_1 \circ (\calX_2 \circ ( \ldots (\calX_{n-1} \circ \calX_n) \ldots ) ). \]
The embeddings field of the CoproductInfo of the resulting crossed module \(\calY\) contains the \(n\) morphisms \(\epsilon_i : \calX_i \to \calY (1 \leqslant i \leqslant n)\).
gap> Y := CoproductXMod( [ XAq8, XAq8, Z8, Z8 ] ); [Group( [ f1, f2, f3, f4, f5, f6, f7, f8 ] )->s4b] gap> StructureDescription( Y ); [ "C2 x C2 x C2 x C2 x C2 x C2 x C2 x C2", "S4" ] gap> CoproductInfo( Y ); rec( embeddings := [ [XAq8 => [Group( [ f1, f2, f3, f4, f5, f6, f7, f8 ] ) -> s4b]], [XAq8 => [Group( [ f1, f2, f3, f4, f5, f6, f7, f8 ] ) -> s4b]], [Z8 => [Group( [ f1, f2, f3, f4, f5, f6, f7, f8 ] ) -> s4b]], [Z8 => [Group( [ f1, f2, f3, f4, f5, f6, f7, f8 ] ) -> s4b]] ], xmods := [ XAq8, XAq8, Z8, Z8 ] )
‣ InducedXMod( args ) | ( function ) |
‣ IsInducedXMod( xmod ) | ( property ) |
‣ InducedXModBySurjection( xmod, hom ) | ( operation ) |
‣ InducedXModByCopower( xmod, hom, list ) | ( operation ) |
‣ MorphismOfInducedXMod( xmod ) | ( attribute ) |
A morphism of crossed modules \((\sigma, \rho) : \calX_1 \to \calX_2\) factors uniquely through an induced crossed module \(\rho_{\ast} \calX_1 = (\delta : \rho_{\ast} S_1 \to R_2)\). Similarly, a morphism of cat\(^1\)-groups factors through an induced cat\(^1\)-group. Calculation of induced crossed modules of \(\calX\) also provides an algebraic means of determining the homotopy \(2\)-type of homotopy pushouts of the classifying space of \(\calX\). For more background from algebraic topology see references in [BH78], [BW95], [BW96]. Induced crossed modules and induced cat\(^1\)-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 crossed module \(\calX = (\partial : S \to R)\) and a homomorphism \(\iota : R \to Q\). The output from the calculation is a crossed module \(\iota_{\ast}\calX = (\delta : \iota_{\ast}S \to Q)\) together with a morphism of crossed modules \(\calX \to \iota_{\ast}\calX\). When \(\iota\) is a surjection with kernel \(K\) then \(\iota_{\ast}S = S/[K,S]\) where \([K,S]\) is the subgroup of \(S\) generated by elements of the form \(s^{-1}s^k, s \in S, k \in K\) (see [BH78], Prop.9). (For many years, up until June 2018, this manual has stated the result to be \([K,S]\), though the correct quotient had been calculated.) 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 \(\calX\) a central extension crossed module of dihedral groups, \((d_{24} \to d_{12})\), and for \(\iota\) a surjection \(d_{12} \to s_3\) with kernel \(c_2\). The induced crossed module is isomorphic to \((d_{12} \to s_3)\).
gap> a := (6,7,8,9)(10,11,12);; b := (7,9)(11,12);; gap> d24 := Group( [ a, b ] );; gap> SetName( d24, "d24" ); gap> c := (1,2)(3,4,5);; d := (4,5);; gap> d12 := Group( [ c, d ] );; gap> SetName( d12, "d12" ); gap> bdy := GroupHomomorphismByImages( d24, d12, [a,b], [c,d] );; gap> X24 := XModByCentralExtension( bdy ); [d24->d12] gap> e := (13,14,15);; f := (14,15);; gap> s3 := Group( [ e, f ] );; gap> SetName( s3, "s3" );; gap> epi := GroupHomomorphismByImages( d12, s3, [c,d], [e,f] );; gap> iX24 := InducedXModBySurjection( X24, epi ); [d24/ker->s3] gap> Display( iX24 ); Crossed module [d24/ker->s3] :- : Source group d24/ker has generators: [ ( 1,11, 5, 4,10, 8)( 2,12, 6, 3, 9, 7), ( 1, 2)( 3, 4)( 5, 9)( 6,10)( 7,11)( 8,12) ] : Range group s3 has generators: [ (13,14,15), (14,15) ] : Boundary homomorphism maps source generators to: [ (13,14,15), (14,15) ] : Action homomorphism maps range generators to automorphisms: (13,14,15) --> { source gens --> [ ( 1,11, 5, 4,10, 8)( 2,12, 6, 3, 9, 7), ( 1, 6)( 2, 5)( 3, 8)( 4, 7)( 9,10)(11,12) ] } (14,15) --> { source gens --> [ ( 1, 8,10, 4, 5,11)( 2, 7, 9, 3, 6,12), ( 1, 2)( 3, 4)( 5, 9)( 6,10)( 7,11)( 8,12) ] } These 2 automorphisms generate the group of automorphisms. gap> morX24 := MorphismOfInducedXMod( iX24 ); [[d24->d12] => [d24/ker->s3]]
For a second, injective example we take for \(\calX\) the result iX24 of the previous example and for \(\iota\) an inclusion of \(s_3\) in \(s_4\). The resulting source group has size \(96\).
gap> g := (16,17,18);; h := (16,17,18,19);; gap> s4 := Group( [ g, h ] );; gap> SetName( s4, "s4" );; gap> iota := GroupHomomorphismByImages( s3, s4, [e,f], [g^2*h^2,g*h^-1] ); [ (13,14,15), (14,15) ] -> [ (17,18,19), (18,19) ] gap> iiX24 := InducedXModByCopower( iX24, iota, [ ] ); i*([d24/ker->s3]) gap> Size2d( iiX24 ); [ 96, 24 ] gap> StructureDescription( iiX24 ); [ "C2 x GL(2,3)", "S4" ]
For a third example we combine the previous two examples by taking for \(\iota\) the more general case alpha = theta*iota. The resulting jX24 is isomorphic to, but not identical to, iiX24.
gap> alpha := CompositionMapping( iota, epi ); [ (1,2)(3,4,5), (4,5) ] -> [ (17,18,19), (18,19) ] gap> jX24 := InducedXMod( X24, alpha );; gap> StructureDescription( jX24 ); [ "C2 x GL(2,3)", "S4" ]
For a fourth example we use the version InducedXMod(Q,R,S) of this global function, with a normal inclusion crossed module \((S \to R)\) and an inclusion mapping \(R \to Q\). We take \((c_6 \to d_{12})\) as \(\calX\) and the inclusion of \(d_{12}\) in \(d_{24}\) as \(\iota\).
## Section 7.2.1 : Example 4 gap> d12b := Subgroup( d24, [ a^2, b ] );; gap> SetName( d12b, "d12b" ); gap> c6b := Subgroup( d12b, [ a^2 ] );; gap> SetName( c6b, "c6b" ); gap> X12 := InducedXMod( d24, d12b, c6b ); i*([c6b->d12b]) gap> StructureDescription( X12 ); [ "C6 x C6", "D24" ] gap> Display( MorphismOfInducedXMod( X12 ) ); Morphism of crossed modules :- : Source = [c6b->d12b] with generating sets: [ ( 6, 8)( 7, 9)(10,12,11) ] [ ( 6, 8)( 7, 9)(10,12,11), ( 7, 9)(11,12) ] : Range = i*([c6b->d12b]) with generating sets: [ ( 4, 5)( 6, 7)( 8, 9)(10,11)(12,13)(14,15), ( 4, 6, 8)( 5, 7, 9)(10,12,14)(11,13,15), ( 4,10)( 5,11)( 6,12)( 7,13)( 8,14)( 9,15), (1,2,3) ] [ ( 6, 7, 8, 9)(10,11,12), ( 7, 9)(11,12) ] : Source Homomorphism maps source generators to: [ ( 4, 9, 6, 5, 8, 7)(10,15,12,11,14,13) ] : Range Homomorphism maps range generators to: [ ( 6, 8)( 7, 9)(10,12,11), ( 7, 9)(11,12) ] #
‣ AllInducedXMods( Q ) | ( operation ) |
This function calculates all the induced crossed modules InducedXMod(Q,R,S), where R runs over all conjugacy classes of subgroups of Q and S runs over all non-trivial normal subgroups of R.
gap> all := AllInducedXMods( q8 );; gap> L := List( all, x -> Source( x ) );; gap> Sort( L, function(g,h) return Size(g) < Size(h); end );; gap> List( L, x -> StructureDescription( x ) ); [ "1", "1", "1", "1", "C2 x C2", "C2 x C2", "C2 x C2", "C4 x C4", "C4 x C4", "C4 x C4", "C2 x C2 x C2 x C2" ]
‣ InducedCat1Group( args ) | ( function ) |
‣ InducedCat1GroupByFreeProduct( grp, hom ) | ( property ) |
This area awaits development.
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