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  7 Induced constructions
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  Before  describing  general  functions  for computing induced structures, we
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  consider coproducts of crossed modules which provide induced crossed modules
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  in certain cases.
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  7.1 Coproducts of crossed modules
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  Need to add here a reference (or two) for coproducts.
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  7.1-1 CoproductXMod
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  CoproductXMod( X1, X2 )  operation
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  This  function  calculates  the  coproduct crossed module of crossed modules
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  mathcal X_1 = (∂_1 : S_1 -> R) and mathcal X_2 = (∂_2 : S_2 -> R) which have
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  a  common range R. The source S_2 of mathcal X_2 acts on S_1 via ∂_2 and the
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  action of mathcal X_1, so we can form a precrossed module (∂' : S_1 ⋉ S_2 ->
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  R)  where  ∂'(s_1,s_2)  =  (∂_1 s_1)(∂_2 s_2). The action of this precrossed
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  module  is the diagonal action (s_1,s_2)^r = (s_1^r,s_2^r). Factoring out by
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  the  Peiffer  subgroup, we obtain the coproduct crossed module mathcal X_1 ∘
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  mathcal X_2.
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  In  the  example  the  structure  descriptions of the precrossed module, the
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  Peiffer   subgroup,  and  the  resulting  coproduct  are  printed  out  when
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  InfoLevel(InfoXMod}  is  at  least  1.  The  coproduct  comes  supplied with
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  attribute  CoproductInfo,  which includes the embedding morphisms of the two
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  factors.
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    Example  
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    gap> q8 := Group( (1,2,3,4)(5,8,7,6), (1,5,3,7)(2,6,4,8) );;
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    gap> X8 := XModByAutomorphismGroup( q8 );;
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    gap> s4 := Range( X8 );; 
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    gap> SetName( q8, "q8" );  SetName( s4, "s4" ); 
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    gap> k4 := NormalSubgroups( s4 )[3];;  SetName( k4, "k4" );
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    gap> Z8 := XModByNormalSubgroup( s4, k4 );;
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    gap> SetName( X8, "X8" );  SetName( Z8, "Z8" );  
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    gap> SetInfoLevel( InfoXMod, 1 ); 
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    gap> XZ8 := CoproductXMod( X8, Z8 );
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    #I  prexmod is [ "C2 x C2 x Q8", "S4" ]
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    #I  peiffer subgroup is C2
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    #I  the coproduct is [ "C2 x C2 x C2 x C2", "S4" ]
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    [Group( [ f1, f2, f3, f4 ] )->s4]
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    gap> SetName( XZ8, "XZ8" ); 
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    gap> info := CoproductInfo( XZ8 );
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    rec( embeddings := [ [X8 => XZ8], [Z8 => XZ8] ], xmods := [ X8, Z8 ] )
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  7.2 Induced crossed modules
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  7.2-1 InducedXMod
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  InducedXMod( args )  function
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  InducedCat1( args )  function
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  IsInducedXMod( xmod )  property
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  MorphismOfInducedXMod( xmod )  attribute
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  A  morphism  of  crossed modules (σ, ρ) : mathcal X_1 -> mathcal X_2 factors
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  uniquely through an induced crossed module ρ_∗ mathcal X_1 = (δ : ρ_∗ S_1 ->
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  R_2).  Similarly,  a  morphism  of  cat1-groups  factors  through an induced
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  cat1-group.  Calculation  of  induced  crossed  modules  of  mathcal  X also
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  provides  an  algebraic means of determining the homotopy 2-type of homotopy
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  pushouts  of  the  classifying  space of mathcal X. For more background from
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  algebraic topology see references in [BH78], [BW95], [BW96]. Induced crossed
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  modules  and  induced  cat1-groups  also  provide  the  building  blocks for
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  constructing pushouts in the categories XMod and Cat1.
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  Data  for  the  cases  of  algebraic  interest  is provided by a conjugation
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  crossed  module  mathcal  X  = (∂ : S -> R) and a homomorphism ι from R to a
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  third  group Q. (It is hoped to implement more general cases in due course.)
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  The output from the calculation is a crossed module ι_∗mathcal X = (δ : ι_∗S
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  -> Q) together with a morphism of crossed modules mathcal X -> ι_∗mathcal X.
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  When  ι is a surjection with kernel K then ι_∗S = [S,K] (see [BH78]). When ι
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  is an inclusion the induced crossed module may be calculated using a copower
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  construction [BW95] or, in the case when R is normal in Q, as a coproduct of
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  crossed  modules  ([BW96],  but  not  yet  implemented). When ι is neither a
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  surjection  nor  an  inclusion,  ι  is  factored  as  the  composite  of the
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  surjection  onto the image and the inclusion of the image in Q, and then the
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  composite  induced  crossed  module  is constructed. These constructions use
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  Tietze transformation routines in the library file tietze.gi.
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  As  a  first, surjective example, we take for mathcal X the normal inclusion
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  crossed  module  of  a4  in  s4, and for ι the surjection from s4 to s3 with
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  kernel k4. The induced crossed module is isomorphic to X3.
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    Example  
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    gap> s4gens := GeneratorsOfGroup( s4 );
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    [ (1,2), (2,3), (3,4) ]
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    gap> a4gens := GeneratorsOfGroup( a4 );
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    [ (1,2,3), (2,3,4) ]
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    gap> s3b := Group( (5,6),(6,7) );;  SetName( s3b, "s3b" );
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    gap> epi := GroupHomomorphismByImages( s4, s3b, s4gens, [(5,6),(6,7),(5,6)] );;
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    gap> X4 := XModByNormalSubgroup( s4, a4 );;
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    gap> indX4 := SurjectiveInducedXMod( X4, epi );
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    [a4/ker->s3b]
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    gap> Display( indX4 );
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    Crossed module [a4/ker->s3b] :- 
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    : Source group a4/ker has generators:
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      [ (1,3,2), (1,2,3) ]
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    : Range group s3b has generators:
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      [ (5,6), (6,7) ]
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    : Boundary homomorphism maps source generators to:
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      [ (5,6,7), (5,7,6) ]
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    : Action homomorphism maps range generators to automorphisms:
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      (5,6) --> { source gens --> [ (1,2,3), (1,3,2) ] }
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      (6,7) --> { source gens --> [ (1,2,3), (1,3,2) ] }
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      These 2 automorphisms generate the group of automorphisms.
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    gap> morX4 := MorphismOfInducedXMod( indX4 );
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    [[a4->s4] => [a4/ker->s3b]]
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  For  a second, injective example we take for mathcal X a conjugation crossed
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  module.
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    Example  
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    gap> d8 := Subgroup( d16, [ b1^2, b2 ] );  SetName( d8, "d8" ); 
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    Group([ (11,13,15,17)(12,14,16,18), (12,18)(13,17)(14,16) ])
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    gap> c4 := Subgroup( d8, [ b1^2 ] );  SetName( c4, "c4" ); 
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    Group([ (11,13,15,17)(12,14,16,18) ])
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    gap> Y16 := XModByNormalSubgroup( d16, d8 );                   
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    [d8->d16]
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    gap> Y8 := SubXMod( Y16, c4, d8 );            
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    [c4->d8]
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    gap> inc8 := InclusionMorphism2DimensionalDomains( Y16, Y8 ); 
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    [[c4->d8] => [d8->d16]]
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    gap> incd8 := RangeHom( inc8 );;
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    gap> indY8 := InducedXMod( Y8, incd8 );
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    #I induced group has Size: 16
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    #I factor 2 is abelian  with invariants: [ 4, 4 ]
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    i*([c4->d8])
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    gap> morY8 := MorphismOfInducedXMod( indY8 );
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    [[c4->d8] => i*([c4->d8])]
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  For a third example we take the identity mapping on s3c as boundary, and the
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  inclusion  of  s3c  in  s4 as ι. The induced group is a general linear group
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  GL(2,3).
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    Example  
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    gap> s3c := Subgroup( s4, [ (2,3), (3,4) ] );;  
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    gap> SetName( s3c, "s3c" );
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    gap> indXs3c := InducedXMod( s4, s3c, s3c );
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    #I induced group has Size: 48
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    i*([s3c->s3c])
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    gap> StructureDescription( indXs3c );
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    [ "GL(2,3)", "S4" ]
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  7.2-2 AllInducedXMods
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  AllInducedXMods( Q )  operation
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  This  function calculates all the induced crossed modules InducedXMod( Q, P,
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  M  ),  where  P runs over all conjugacy classes of subgroups of Q and M runs
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  over all non-trivial subgroups of P.
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