<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% --> <!-- --> <!-- gp2ind.xml XMOD documentation Chris Wensley --> <!-- --> <!-- $Id: gp2ind.xml,v 2.11 2008/04/30 gap Exp $ --> <!-- --> <!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% --> <?xml version="1.0" encoding="ISO-8859-15"?> <!-- <M>Id: gp2ind.xml,v 2.009 Exp <M> --> <Chapter Label="chap-induce2"> <Heading>Induced Constructions</Heading> <Section><Heading>Induced crossed modules</Heading> <Index>induced crossed module</Index> <ManSection> <Func Name="InducedXMod" Arg="args" /> <Func Name="InducedCat1" Arg="args" /> <Prop Name="IsInducedXMod" Arg="xmod" /> <Prop Name="IsInducedCat1" Arg="cat1" /> <Attr Name="MorphismOfInducedXMod" Arg="xmod" /> <Description> A morphism of crossed modules <M>(\sigma, \rho) : {\cal X}_1 \to {\cal X}_2</M> factors uniquely through an induced crossed module <M>\rho_{\ast} {\cal X}_1 = (\delta ~:~ \rho_{\ast} S_1 \to R_2)</M>. Similarly, a morphism of cat1-groups factors through an induced cat1-group. Calculation of induced crossed modules of <M>{\cal X}</M> also provides an algebraic means of determining the homotopy <M>2</M>-type of homotopy pushouts of the classifying space of <M>{\cal X}</M>. For more background from algebraic topology see references in \cite{BH1}, \cite{BW1}, \cite{BW2}. Induced crossed modules and induced cat1-groups also provide the building blocks for constructing pushouts in the categories <E>XMod</E> and <E>Cat1</E>. <P/> Data for the cases of algebraic interest is provided by a conjugation crossed module <M>{\cal X} = (\partial ~:~ S \to R)</M> and a homomorphism <M>\iota</M> from <M>R</M> to a third group <M>Q</M>. The output from the calculation is a crossed module <M>\iota_{\ast}{\cal X} = (\delta ~:~ \iota_{\ast}S \to Q)</M> together with a morphism of crossed modules <M>{\cal X} \to \iota_{\ast}{\cal X}</M>. When <M>\iota</M> is a surjection with kernel <M>K</M> then <M>\iota_{\ast}S = [S,K]</M> (see \cite{BH1}). When <M>\iota</M> is an inclusion the induced crossed module may be calculated using a copower construction \cite{BW1} or, in the case when <M>R</M> is normal in <M>Q</M>, as a coproduct of crossed modules (\cite{BW2}, but not yet implemented). When <M>\iota</M> is neither a surjection nor an inclusion, <M>\iota</M> is written as the composite of the surjection onto the image and the inclusion of the image in <M>Q</M>, and then the composite induced crossed module is constructed. These constructions use Tietze transformation routines in the library file <C>tietze.gi</C>. <P/> As a first, surjective example, we take for <M>{\cal X}</M> the normal inclusion crossed module of <C>a4</C> in <C>s4</C>, and for <M>\iota</M> the surjection from <C>s4</C> to <C>s3</C> with kernel <C>k4</C>. The induced crossed module is isomorphic to <C>X3</C>. </Description> </ManSection> <Example> <![CDATA[ gap> s4gens := [ (1,2), (2,3), (3,4) ];; gap> s4 := Group( s4gens );; SetName(s4,"s4"); gap> a4gens := [ (1,2,3), (2,3,4) ];; gap> a4 := Subgroup( s4, a4gens );; SetName( a4, "a4" ); gap> s3 := Group( (5,6),(6,7) );; SetName( s3, "s3" ); gap> epi := GroupHomomorphismByImages( s4, s3, s4gens, [(5,6),(6,7),(5,6)] );; gap> X4 := XModByNormalSubgroup( s4, a4 );; gap> indX4 := SurjectiveInducedXMod( X4, epi ); [a4/ker->s3] gap> morX4 := MorphismOfInducedXMod( indX4 ); [[a4->s4] => [a4/ker->s3]] ]]> </Example> For a second, injective example we take for <M>{\cal X}</M> the conjugation crossed module <M>(\partial ~:~ c4 \to d8)</M> of Chapter 3, and for <M>\iota</M> the inclusion <C>incd8</C> of <C>d8</C> in <C>d16</C>. The induced crossed module has <M>c4 \times c4</M> as source. <Example> <![CDATA[ gap> incd8 := RangeHom( inc8 );; gap> [ Source(incd8), Range(incd8), IsInjective(incd8) ]; [ d8, d16, true ] gap> indX8 := InducedXMod( X8, incd8 ); #I Simplified presentation for induced group :- <presentation with 2 gens and 3 rels of total length 12> #I generators: [ f11, f14 ] #I relators: #I 1. 4 [ 1, 1, 1, 1 ] #I 2. 4 [ 2, 2, 2, 2 ] #I 3. 4 [ 2, -1, -2, 1 ] #I induced group has Size: 16 #I factor 1 is abelian with invariants: [ 4, 4 ] i*([c4->d8]) gap> Display( indX8 ); Crossed module i*([c4->d8]) :- : Source group has generators: [ ( 1, 2, 6, 3)( 4, 7,12, 9)( 5, 8,13,10)(11,14,16,15), ( 1, 4,11, 5)( 2, 7,14, 8)( 3, 9,15,10)( 6,12,16,13) ] : Range group d16 has generators: [ (11,12,13,14,15,16,17,18), (12,18)(13,17)(14,16) ] : Boundary homomorphism maps source generators to: [ (11,13,15,17)(12,14,16,18), (11,17,15,13)(12,18,16,14) ] : Action homomorphism maps range generators to automorphisms: (11,12,13,14,15,16,17,18) --> { source gens --> [ ( 1, 5,11, 4)( 2, 8,14, 7)( 3,10,15, 9)( 6,13,16,12), ( 1, 3, 6, 2)( 4, 9,12, 7)( 5,10,13, 8)(11,15,16,14) ] } (12,18)(13,17)(14,16) --> { source gens --> [ ( 1, 3, 6, 2)( 4, 9,12, 7)( 5,10,13, 8)(11,15,16,14), ( 1, 5,11, 4)( 2, 8,14, 7)( 3,10,15, 9)( 6,13,16,12) ] } These 2 automorphisms generate the group of automorphisms. gap> morX8 := MorphismOfInducedXMod( indX8 ); [[c4->d8] => i*([c4->d8])] gap> Display( morX8 ); Morphism of crossed modules :- : Source = [c4->d8] with generating sets: [ (11,13,15,17)(12,14,16,18) ] [ (11,13,15,17)(12,14,16,18), (12,18)(13,17)(14,16) ] : Range = i*([c4->d8]) with generating sets: [ ( 1, 2, 6, 3)( 4, 7,12, 9)( 5, 8,13,10)(11,14,16,15), ( 1, 4,11, 5)( 2, 7,14, 8)( 3, 9,15,10)( 6,12,16,13) ] [ (11,12,13,14,15,16,17,18), (12,18)(13,17)(14,16) ] : Source Homomorphism maps source generators to: [ ( 1, 2, 6, 3)( 4, 7,12, 9)( 5, 8,13,10)(11,14,16,15) ] : Range Homomorphism maps range generators to: [ (11,13,15,17)(12,14,16,18), (12,18)(13,17)(14,16) ] ]]> </Example> For a third example we take the identity mapping on <C>s3</C> as boundary, and the inclusion of <C>s3</C> in <C>s4</C> as <C>iota</C>. The induced group is a general linear group <C>GL(2,3)</C>. <Example> <![CDATA[ gap> s3b := Subgroup( s4, [ (2,3), (3,4) ] );; SetName( s3b, "s3b" ); gap> indX3 := InducedXMod( s4, s3b, s3b ); #I Simplified presentation for induced group :- <presentation with 2 gens and 4 rels of total length 33> #I generators: [ f11, f112 ] #I relators: #I 1. 2 [ 1, 1 ] #I 2. 3 [ 2, 2, 2 ] #I 3. 12 [ 1, -2, 1, 2, 1, 2, 1, -2, 1, 2, 1, 2 ] #I 4. 16 [ -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1 ] #I induced group has Size: 48 #I IdGroup = [ [ 48, 29 ] ] i*([s3b->s3b]) gap> isoX3 := IsomorphismGroups( Source( indX3 ), GeneralLinearGroup(2,3) ); [ (1,2)(4,5)(6,8), (2,3,4)(5,6,7) ] -> [ [ [ Z(3)^0, 0*Z(3) ], [ Z(3), Z(3) ] ], [ [ Z(3)^0, Z(3)^0 ], [ 0*Z(3), Z(3)^0 ] ] ] ]]> </Example> <ManSection> <Oper Name="AllInducedXMods" Arg="Q" /> <Description> This function calculates all the induced crossed modules <C>InducedXMod( Q, P, M )</C>, where <C>P</C> runs over all conjugacy classes of subgroups of <C>Q</C> and <C>M</C> runs over all non-trivial subgroups of <C>P</C>. </Description> </ManSection> </Section> </Chapter>