  
  [1X6 [33X[0;0YInduced Constructions[133X[101X
  
  
  [1X6.1 [33X[0;0YInduced crossed modules[133X[101X
  
  [1X6.1-1 InducedXMod[101X
  
  [29X[2XInducedXMod[102X( [3Xargs[103X ) [32X function
  [29X[2XInducedCat1[102X( [3Xargs[103X ) [32X function
  [29X[2XIsInducedXMod[102X( [3Xxmod[103X ) [32X property
  [29X[2XIsInducedCat1[102X( [3Xcat1[103X ) [32X property
  [29X[2XMorphismOfInducedXMod[102X( [3Xxmod[103X ) [32X attribute
  
  [33X[0;0YA  morphism  of crossed modules [22X(σ, ρ) : cal X_1 -> cal X_2[122X factors uniquely
  through  an  induced  crossed  module  [22Xρ_∗  cal  X_1 = (δ : ρ_∗ S_1 -> R_2)[122X.
  Similarly,  a morphism of cat1-groups factors through an induced cat1-group.
  Calculation  of  induced crossed modules of [22Xcal X[122X also provides an algebraic
  means  of  determining  the  homotopy  [22X2[122X-type  of  homotopy  pushouts of the
  classifying  space of [22Xcal X[122X. 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 [13XXMod[113X and [13XCat1[113X.[133X
  
  [33X[0;0YData  for  the  cases  of  algebraic  interest  is provided by a conjugation
  crossed  module  [22Xcal X = (∂ : S -> R)[122X and a homomorphism [22Xι[122X from [22XR[122X to a third
  group [22XQ[122X. The output from the calculation is a crossed module [22Xι_∗cal X = (δ :
  ι_∗S  ->  Q)[122X  together with a morphism of crossed modules [22Xcal X -> ι_∗cal X[122X.
  When  [22Xι[122X is a surjection with kernel [22XK[122X then [22Xι_∗S = [S,K][122X (see [BH78]). When [22Xι[122X
  is an inclusion the induced crossed module may be calculated using a copower
  construction [BW95] or, in the case when [22XR[122X is normal in [22XQ[122X, as a coproduct of
  crossed  modules  ([BW96],  but  not  yet  implemented). When [22Xι[122X is neither a
  surjection nor an inclusion, [22Xι[122X is written as the composite of the surjection
  onto  the  image and the inclusion of the image in [22XQ[122X, and then the composite
  induced  crossed  module  is  constructed.  These  constructions  use Tietze
  transformation routines in the library file [10Xtietze.gi[110X.[133X
  
  [33X[0;0YAs  a  first,  surjective  example,  we  take for [22Xcal X[122X the normal inclusion
  crossed  module  of  [10Xa4[110X  in  [10Xs4[110X, and for [22Xι[122X the surjection from [10Xs4[110X to [10Xs3[110X with
  kernel [10Xk4[110X. The induced crossed module is isomorphic to [10XX3[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xs4gens := [ (1,2), (2,3), (3,4) ];;[127X[104X
    [4X[25Xgap>[125X [27Xs4 := Group( s4gens );; SetName(s4,"s4");[127X[104X
    [4X[25Xgap>[125X [27Xa4gens := [ (1,2,3), (2,3,4) ];;[127X[104X
    [4X[25Xgap>[125X [27Xa4 := Subgroup( s4, a4gens );;  SetName( a4, "a4" );[127X[104X
    [4X[25Xgap>[125X [27Xs3 := Group( (5,6),(6,7) );;  SetName( s3, "s3" );[127X[104X
    [4X[25Xgap>[125X [27Xepi := GroupHomomorphismByImages( s4, s3, s4gens, [(5,6),(6,7),(5,6)] );;[127X[104X
    [4X[25Xgap>[125X [27XX4 := XModByNormalSubgroup( s4, a4 );;[127X[104X
    [4X[25Xgap>[125X [27XindX4 := SurjectiveInducedXMod( X4, epi );[127X[104X
    [4X[28X[a4/ker->s3][128X[104X
    [4X[25Xgap>[125X [27XDisplay( indX4 );[127X[104X
    [4X[28X[128X[104X
    [4X[28XCrossed module [a4/ker->s3] :- [128X[104X
    [4X[28X: Source group a4/ker has generators:[128X[104X
    [4X[28X  [ (1,3,2), (1,2,3) ][128X[104X
    [4X[28X: Range group s3 has generators:[128X[104X
    [4X[28X  [ (5,6), (6,7) ][128X[104X
    [4X[28X: Boundary homomorphism maps source generators to:[128X[104X
    [4X[28X  [ (5,6,7), (5,7,6) ][128X[104X
    [4X[28X: Action homomorphism maps range generators to automorphisms:[128X[104X
    [4X[28X  (5,6) --> { source gens --> [ (1,2,3), (1,3,2) ] }[128X[104X
    [4X[28X  (6,7) --> { source gens --> [ (1,2,3), (1,3,2) ] }[128X[104X
    [4X[28X  These 2 automorphisms generate the group of automorphisms.[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XmorX4 := MorphismOfInducedXMod( indX4 );[127X[104X
    [4X[28X[[a4->s4] => [a4/ker->s3]][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor  a  second,  injective example we take for [22Xcal X[122X the conjugation crossed
  module  [22X(∂  : c4 -> d8)[122X of Chapter 3, and for [22Xι[122X the inclusion [10Xincd8[110X of [10Xd8[110X in
  [10Xd16[110X.  The induced crossed module has [22Xc4 × c4[122X as source. (The groups in [10XindX8[110X
  vary from run to run due to random methods.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xincd8 := RangeHom( inc8 );;[127X[104X
    [4X[25Xgap>[125X [27X[ Source(incd8), Range(incd8), IsInjective(incd8) ];[127X[104X
    [4X[28X[ d8, d16, true ][128X[104X
    [4X[25Xgap>[125X [27XindX8 := InducedXMod( X8, incd8 );[127X[104X
    [4X[28X#I induced group has Size: 16[128X[104X
    [4X[28X#I factor 2 is abelian  with invariants: [ 4, 4 ][128X[104X
    [4X[28Xi*([c4->d8])[128X[104X
    [4X[25Xgap>[125X [27XDisplay( indX8 ); [127X[104X
    [4X[28X[128X[104X
    [4X[28XCrossed module i*([c4->d8]) :- [128X[104X
    [4X[28X: Source group has generators:[128X[104X
    [4X[28X  [ ( 1, 2, 6, 3)( 4, 7,12, 9)( 5, 8,13,10)(11,14,16,15), [128X[104X
    [4X[28X  ( 1, 4,11, 5)( 2, 7,14, 8)( 3, 9,15,10)( 6,12,16,13) ][128X[104X
    [4X[28X: Range group d16 has generators:[128X[104X
    [4X[28X  [ (11,12,13,14,15,16,17,18), (12,18)(13,17)(14,16) ][128X[104X
    [4X[28X: Boundary homomorphism maps source generators to:[128X[104X
    [4X[28X  [ (11,13,15,17)(12,14,16,18), (11,17,15,13)(12,18,16,14) ][128X[104X
    [4X[28X: Action homomorphism maps range generators to automorphisms:[128X[104X
    [4X[28X  (11,12,13,14,15,16,17,18) --> { source gens --> [128X[104X
    [4X[28X[ ( 1, 5,11, 4)( 2, 8,14, 7)( 3,10,15, 9)( 6,13,16,12), [128X[104X
    [4X[28X  ( 1, 3, 6, 2)( 4, 9,12, 7)( 5,10,13, 8)(11,15,16,14) ] }[128X[104X
    [4X[28X  (12,18)(13,17)(14,16) --> { source gens --> [128X[104X
    [4X[28X[ ( 1, 3, 6, 2)( 4, 9,12, 7)( 5,10,13, 8)(11,15,16,14), [128X[104X
    [4X[28X  ( 1, 5,11, 4)( 2, 8,14, 7)( 3,10,15, 9)( 6,13,16,12) ] }[128X[104X
    [4X[28X  These 2 automorphisms generate the group of automorphisms.[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XmorX8 := MorphismOfInducedXMod( indX8 );[127X[104X
    [4X[28X[[c4->d8] => i*([c4->d8])][128X[104X
    [4X[25Xgap>[125X [27XDisplay( morX8 ); [127X[104X
    [4X[28XMorphism of crossed modules :- [128X[104X
    [4X[28X: Source = [c4->d8] with generating sets:[128X[104X
    [4X[28X  [ (11,13,15,17)(12,14,16,18) ][128X[104X
    [4X[28X  [ (11,13,15,17)(12,14,16,18), (12,18)(13,17)(14,16) ][128X[104X
    [4X[28X:  Range = i*([c4->d8]) with generating sets:[128X[104X
    [4X[28X  [ ( 1, 2, 6, 3)( 4, 7,12, 9)( 5, 8,13,10)(11,14,16,15), [128X[104X
    [4X[28X  ( 1, 4,11, 5)( 2, 7,14, 8)( 3, 9,15,10)( 6,12,16,13) ][128X[104X
    [4X[28X  [ (11,12,13,14,15,16,17,18), (12,18)(13,17)(14,16) ][128X[104X
    [4X[28X: Source Homomorphism maps source generators to:[128X[104X
    [4X[28X  [ ( 1, 2, 6, 3)( 4, 7,12, 9)( 5, 8,13,10)(11,14,16,15) ][128X[104X
    [4X[28X: Range Homomorphism maps range generators to:[128X[104X
    [4X[28X  [ (11,13,15,17)(12,14,16,18), (12,18)(13,17)(14,16) ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor  a third example we take the identity mapping on [10Xs3[110X as boundary, and the
  inclusion  of  [10Xs3[110X in [10Xs4[110X as [10Xiota[110X. The induced group is a general linear group
  [10XGL(2,3)[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xs3b := Subgroup( s4, [ (2,3), (3,4) ] );;  [127X[104X
    [4X[25Xgap>[125X [27XSetName( s3b, "s3b" );[127X[104X
    [4X[25Xgap>[125X [27XindX3 := InducedXMod( s4, s3b, s3b );[127X[104X
    [4X[28X#I induced group has Size: 48[128X[104X
    [4X[28Xi*([s3b->s3b])[128X[104X
    [4X[25Xgap>[125X [27XisoX3 := IsomorphismGroups( Source( indX3 ), GeneralLinearGroup(2,3) );[127X[104X
    [4X[28X[ (2,3)(4,7)(5,6), (1,2,3)(4,8,7), (2,4,5)(3,6,7) ] -> [128X[104X
    [4X[28X[ [ [ 0*Z(3), Z(3)^0 ], [ Z(3)^0, 0*Z(3) ] ], [128X[104X
    [4X[28X  [ [ Z(3), Z(3) ], [ Z(3)^0, 0*Z(3) ] ], [128X[104X
    [4X[28X  [ [ Z(3), Z(3)^0 ], [ Z(3), 0*Z(3) ] ] ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X6.1-2 AllInducedXMods[101X
  
  [29X[2XAllInducedXMods[102X( [3XQ[103X ) [32X operation
  
  [33X[0;0YThis  function calculates all the induced crossed modules [10XInducedXMod( Q, P,
  M  )[110X,  where  [10XP[110X runs over all conjugacy classes of subgroups of [10XQ[110X and [10XM[110X runs
  over all non-trivial subgroups of [10XP[110X.[133X
  
