  
  [1X3 [33X[0;0Y2d-mappings[133X[101X
  
  
  [1X3.1 [33X[0;0YMorphisms of 2-dimensional groups[133X[101X
  
  [33X[0;0YThis    chapter   describes   morphisms   of   (pre-)crossed   modules   and
  (pre-)cat1-groups.[133X
  
  [1X3.1-1 Source[101X
  
  [33X[1;0Y[29X[2XSource[102X( [3Xmap[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XRange[102X( [3Xmap[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XSourceHom[102X( [3Xmap[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XRangeHom[102X( [3Xmap[103X ) [32X attribute[133X
  
  [33X[0;0YMorphisms of [13X2-dimensional groups[113X are implemented as [13X2-dimensional mappings[113X.
  These have a pair of 2-dimensional groups as source and range, together with
  two  group  homomorphisms  mapping  between  corresponding  source and range
  groups. These functions return [10Xfail[110X when invalid data is supplied.[133X
  
  
  [1X3.2 [33X[0;0YMorphisms of pre-crossed modules[133X[101X
  
  [1X3.2-1 IsXModMorphism[101X
  
  [33X[1;0Y[29X[2XIsXModMorphism[102X( [3Xmap[103X ) [32X property[133X
  [33X[1;0Y[29X[2XIsPreXModMorphism[102X( [3Xmap[103X ) [32X property[133X
  
  [33X[0;0YA  morphism  between two pre-crossed modules [22XmathcalX_1 = (∂_1 : S_1 -> R_1)[122X
  and  [22XmathcalX_2  = (∂_2 : S_2 -> R_2)[122X is a pair [22X(σ, ρ)[122X, where [22Xσ : S_1 -> S_2[122X
  and  [22Xρ : R_1 -> R_2[122X commute with the two boundary maps and are morphisms for
  the two actions:[133X
  
  
  [24X[33X[0;6Y\partial_2  \circ  \sigma  ~=~ \rho \circ \partial_1, \qquad \sigma(s^r) ~=~
  (\sigma s)^{\rho r}.[133X
  
  [124X
  
  [33X[0;0YHere  [22Xσ[122X  is  the  [2XSourceHom[102X  ([14X3.1-1[114X)  and  [22Xρ[122X  is the [2XRangeHom[102X ([14X3.1-1[114X) of the
  morphism.  When  [22XmathcalX_1 = mathcalX_2[122X and [22Xσ, ρ[122X are automorphisms then [22X(σ,
  ρ)[122X  is  an automorphism of [22XmathcalX_1[122X. The group of automorphisms is denoted
  by [22XAut(mathcalX_1 )[122X.[133X
  
  [1X3.2-2 IsInjective[101X
  
  [33X[1;0Y[29X[2XIsInjective[102X( [3Xmap[103X ) [32X method[133X
  [33X[1;0Y[29X[2XIsSurjective[102X( [3Xmap[103X ) [32X method[133X
  [33X[1;0Y[29X[2XIsSingleValued[102X( [3Xmap[103X ) [32X method[133X
  [33X[1;0Y[29X[2XIsTotal[102X( [3Xmap[103X ) [32X method[133X
  [33X[1;0Y[29X[2XIsBijective[102X( [3Xmap[103X ) [32X method[133X
  [33X[1;0Y[29X[2XIsEndo2DimensionalMapping[102X( [3Xmap[103X ) [32X property[133X
  
  [33X[0;0YThe  usual  properties  of  mappings  are  easily  checked.  It  is  usually
  sufficient  to  verify  that  both  the  [2XSourceHom[102X  ([14X3.1-1[114X) and the [2XRangeHom[102X
  ([14X3.1-1[114X) have the required property.[133X
  
  [1X3.2-3 XModMorphism[101X
  
  [33X[1;0Y[29X[2XXModMorphism[102X( [3Xargs[103X ) [32X function[133X
  [33X[1;0Y[29X[2XXModMorphismByHoms[102X( [3XX1[103X, [3XX2[103X, [3Xsigma[103X, [3Xrho[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XPreXModMorphism[102X( [3Xargs[103X ) [32X function[133X
  [33X[1;0Y[29X[2XPreXModMorphismByHoms[102X( [3XP1[103X, [3XP2[103X, [3Xsigma[103X, [3Xrho[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XInclusionMorphism2DimensionalDomains[102X( [3XX1[103X, [3XS1[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XInnerAutomorphismXMod[102X( [3XX1[103X, [3Xr[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XIdentityMapping[102X( [3XX1[103X ) [32X attribute[133X
  
  [33X[0;0YThese are the constructors for morphisms of pre-crossed and crossed modules.[133X
  
  [33X[0;0YIn  the  following example we construct a simple automorphism of the crossed
  module [10XX1[110X constructed in the previous chapter.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xsigma1 := GroupHomomorphismByImages( c5, c5, [ (5,6,7,8,9) ][127X[104X
    [4X[28X        [ (5,9,8,7,6) ] );;[128X[104X
    [4X[25Xgap>[125X [27Xrho1 := IdentityMapping( Range( X1 ) );[127X[104X
    [4X[28XIdentityMapping( PAut(c5) )[128X[104X
    [4X[25Xgap>[125X [27Xmor1 := XModMorphism( X1, X1, sigma1, rho1 );[127X[104X
    [4X[28X[[c5->Aut(c5))] => [c5->Aut(c5))]] [128X[104X
    [4X[25Xgap>[125X [27XDisplay( mor1 );[127X[104X
    [4X[28XMorphism of crossed modules :- [128X[104X
    [4X[28X: Source = [c5->Aut(c5)] with generating sets:[128X[104X
    [4X[28X  [ (5,6,7,8,9) ][128X[104X
    [4X[28X  [ GroupHomomorphismByImages( c5, c5, [ (5,6,7,8,9) ], [ (5,7,9,6,8) ] ) ][128X[104X
    [4X[28X: Range = Source[128X[104X
    [4X[28X: Source Homomorphism maps source generators to:[128X[104X
    [4X[28X  [ (5,9,8,7,6) ][128X[104X
    [4X[28X: Range Homomorphism maps range generators to:[128X[104X
    [4X[28X  [ GroupHomomorphismByImages( c5, c5, [ (5,6,7,8,9) ], [ (5,7,9,6,8) ] ) ][128X[104X
    [4X[25Xgap>[125X [27XIsAutomorphism2DimensionalDomain( mor1 );[127X[104X
    [4X[28Xtrue [128X[104X
    [4X[25Xgap>[125X [27XOrder( mor1 );[127X[104X
    [4X[28X2[128X[104X
    [4X[25Xgap>[125X [27XRepresentationsOfObject( mor1 );[127X[104X
    [4X[28X[ "IsComponentObjectRep", "IsAttributeStoringRep", "Is2DimensionalMappingRep" ][128X[104X
    [4X[25Xgap>[125X [27XKnownPropertiesOfObject( mor1 );[127X[104X
    [4X[28X[ "CanEasilyCompareElements", "CanEasilySortElements", "IsTotal", [128X[104X
    [4X[28X  "IsSingleValued", "IsInjective", "IsSurjective", "RespectsMultiplication", [128X[104X
    [4X[28X  "IsPreXModMorphism", "IsXModMorphism", "IsEndomorphism2DimensionalDomain", [128X[104X
    [4X[28X  "IsAutomorphism2DimensionalDomain" ][128X[104X
    [4X[25Xgap>[125X [27XKnownAttributesOfObject( mor1 );[127X[104X
    [4X[28X[ "Name", "Order", "Range", "Source", "SourceHom", "RangeHom" ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X3.2-4 IsomorphismPerm2DimensionalGroup[101X
  
  [33X[1;0Y[29X[2XIsomorphismPerm2DimensionalGroup[102X( [3Xobj[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XIsomorphismPc2DimensionalGroup[102X( [3Xobj[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XIsomorphismByIsomorphisms[102X( [3XD[103X, [3Xlist[103X ) [32X operation[133X
  
  [33X[0;0YWhen [22Xmathcal D[122X is a [22X2[122X-dimensional domain with source [22XS[122X and range [22XR[122X and [22Xσ : S
  ->     S',~     ρ     :     R     ->     R'[122X     are    isomorphisms,    then
  [10XIsomorphismByIsomorphisms(D,[sigma,rho])[110X  returns  an  isomorphism  [22X(σ,ρ)  :
  mathcal D -> mathcal D'[122X where [22Xmathcal D'[122X has source [22XS'[122X and range [22XR'[122X. Be sure
  to  test  [10XIsBijective[110X  for  the  two  functions  [22Xσ,ρ[122X  before  applying  this
  operation.[133X
  
  [33X[0;0YUsing  [2XIsomorphismByIsomorphisms[102X  with a pair of isomorphisms obtained using
  [10XIsomorphismPermGroup[110X  or  [10XIsomorphismPcGroup[110X,  we  may  construct  a crossed
  module or a cat1-group of permutation groups or pc-groups.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xq8 := SmallGroup(8,4);;   ## quaternion group [127X[104X
    [4X[25Xgap>[125X [27XXq8 := XModByAutomorphismGroup( q8 );[127X[104X
    [4X[28X[Group( [ f1, f2, f3 ] )->Group( [ f1, f2, f3, f4 ] )][128X[104X
    [4X[25Xgap>[125X [27Xiso := IsomorphismPerm2DimensionalGroup( Xq8 );;[127X[104X
    [4X[25Xgap>[125X [27XYq8 := Image( iso );[127X[104X
    [4X[28X[Group( [ (1,2,4,6)(3,8,7,5), (1,3,4,7)(2,5,6,8), (1,4)(2,6)(3,7)(5,8) [128X[104X
    [4X[28X ] )->Group( [ (2,6,5,4), (1,2,4)(3,5,6), (2,5)(4,6), (1,3)(2,5) ] )][128X[104X
    [4X[25Xgap>[125X [27Xs4 := SymmetricGroup(4);; [127X[104X
    [4X[25Xgap>[125X [27Xisos4 := IsomorphismGroups( Range(Yq8), s4 );;[127X[104X
    [4X[25Xgap>[125X [27Xid := IdentityMapping( Source( Yq8 ) );; [127X[104X
    [4X[25Xgap>[125X [27XIsBijective( id );;  IsBijective( isos4 );;[127X[104X
    [4X[25Xgap>[125X [27Xmor := IsomorphismByIsomorphisms( Yq8, [id,isos4] );;[127X[104X
    [4X[25Xgap>[125X [27XZq8 := Image( mor );[127X[104X
    [4X[28X[Group( [ (1,2,4,6)(3,8,7,5), (1,3,4,7)(2,5,6,8), (1,4)(2,6)(3,7)(5,8) [128X[104X
    [4X[28X ] )->SymmetricGroup( [ 1 .. 4 ] )][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X3.3 [33X[0;0YMorphisms of pre-cat1-groups[133X[101X
  
  [33X[0;0YA  morphism  of pre-cat1-groups from [22XmathcalC_1 = (e_1;t_1,h_1 : G_1 -> R_1)[122X
  to [22XmathcalC_2 = (e_2;t_2,h_2 : G_2 -> R_2)[122X is a pair [22X(γ, ρ)[122X where [22Xγ : G_1 ->
  G_2[122X and [22Xρ : R_1 -> R_2[122X are homomorphisms satisfying[133X
  
  
  [24X[33X[0;6Yh_2  \circ \gamma ~=~ \rho \circ h_1, \qquad t_2 \circ \gamma ~=~ \rho \circ
  t_1, \qquad e_2 \circ \rho ~=~ \gamma \circ e_1.[133X
  
  [124X
  
  [1X3.3-1 IsCat1Morphism[101X
  
  [33X[1;0Y[29X[2XIsCat1Morphism[102X( [3Xmap[103X ) [32X property[133X
  [33X[1;0Y[29X[2XIsPreCat1Morphism[102X( [3Xmap[103X ) [32X property[133X
  [33X[1;0Y[29X[2XCat1Morphism[102X( [3Xargs[103X ) [32X function[133X
  [33X[1;0Y[29X[2XCat1MorphismByHoms[102X( [3XC1[103X, [3XC2[103X, [3Xgamma[103X, [3Xrho[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XPreCat1Morphism[102X( [3Xargs[103X ) [32X function[133X
  [33X[1;0Y[29X[2XPreCat1MorphismByHoms[102X( [3XP1[103X, [3XP2[103X, [3Xgamma[103X, [3Xrho[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XInclusionMorphism2DimensionalDomains[102X( [3XC1[103X, [3XS1[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XInnerAutomorphismCat1[102X( [3XC1[103X, [3Xr[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XIdentityMapping[102X( [3XC1[103X ) [32X attribute[133X
  
  [33X[0;0YFor  an  example we form a second cat1-group [10XC2=[g18=>s3a][110X, similar to [10XC1[110X in
  [14X2.4-1[114X, then construct an isomorphism [22X(γ,ρ)[122X between them.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xt2 := GroupHomomorphismByImages(g18,s3a,g18gens,[(),(7,8,9),(8,9)]);;     [127X[104X
    [4X[25Xgap>[125X [27Xe2 := GroupHomomorphismByImages(s3a,g18,s3agens,[(4,5,6),(2,3)(5,6)]);;   [127X[104X
    [4X[25Xgap>[125X [27XC2 := Cat1Group( t2, h1, e2 );; [127X[104X
    [4X[25Xgap>[125X [27Ximgamma := [ (4,5,6), (1,2,3), (2,3)(5,6) ];; [127X[104X
    [4X[25Xgap>[125X [27Xgamma := GroupHomomorphismByImages( g18, g18, g18gens, imgamma );;[127X[104X
    [4X[25Xgap>[125X [27Xrho := IdentityMapping( s3a );; [127X[104X
    [4X[25Xgap>[125X [27Xmor := Cat1Morphism( C1, C2, gamma, rho );;[127X[104X
    [4X[25Xgap>[125X [27XDisplay( mor );;[127X[104X
    [4X[28XMorphism of cat1-groups :- [128X[104X
    [4X[28X: Source = [g18=>s3a] with generating sets:[128X[104X
    [4X[28X  [ (1,2,3), (4,5,6), (2,3)(5,6) ][128X[104X
    [4X[28X  [ (7,8,9), (8,9) ][128X[104X
    [4X[28X:  Range = [g18=>s3a] with generating sets:[128X[104X
    [4X[28X  [ (1,2,3), (4,5,6), (2,3)(5,6) ][128X[104X
    [4X[28X  [ (7,8,9), (8,9) ][128X[104X
    [4X[28X: Source Homomorphism maps source generators to:[128X[104X
    [4X[28X  [ (4,5,6), (1,2,3), (2,3)(5,6) ][128X[104X
    [4X[28X: Range Homomorphism maps range generators to:[128X[104X
    [4X[28X  [ (7,8,9), (8,9) ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X3.3-2 IsomorphismPermObject[101X
  
  [33X[1;0Y[29X[2XIsomorphismPermObject[102X( [3Xobj[103X ) [32X function[133X
  [33X[1;0Y[29X[2XIsomorphismPerm2DimensionalGroup[102X( [3X2DimensionalGroup[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XIsomorphismFp2DimensionalGroup[102X( [3X2DimensionalGroup[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XIsomorphismPc2DimensionalGroup[102X( [3X2DimensionalGroup[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XSmallerDegreePerm2DimensionalDomain[102X( [3X2DimensionalDomain[103X ) [32X function[133X
  
  [33X[0;0YThe         global        function        [10XIsomorphismPermObject[110X        calls
  [10XIsomorphismPerm2DimensionalGroup[110X,   which   constructs   a   morphism  whose
  [2XSourceHom[102X    ([14X3.1-1[114X)    and    [2XRangeHom[102X   ([14X3.1-1[114X)   are   calculated   using
  [10XIsomorphismPermGroup[110X  on  the  source  and  range.  Similarly  the operation
  [10XSmallerDegreePermutationRepresentation[110X  may  be  used  on  the two groups to
  obtain the attribute [10XSmallerDegreePerm2DimensionalDomain[110X. Names are assigned
  automatically.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xiso2 := IsomorphismPerm2DimensionalGroup( C2 );[127X[104X
    [4X[28X[[G2=>d12] => [..]][128X[104X
    [4X[25Xgap>[125X [27XDisplay( iso2 );[127X[104X
    [4X[28XMorphism of cat1-groups :- [128X[104X
    [4X[28X: Source = [G2=>d12] with generating sets:[128X[104X
    [4X[28X  [ f1, f2, f3, f4, f5, f6, f7 ][128X[104X
    [4X[28X  [ f1, f2, f3 ][128X[104X
    [4X[28X:  Range = P[G2=>d12] with generating sets:[128X[104X
    [4X[28X  [ ( 6,12)( 8,15)( 9,16)(11,19)(13,26)(14,22)(17,27)(18,25)(20,21)(23,24), [128X[104X
    [4X[28X  ( 2, 3)( 5,10)( 9,16)(11,18)(17,23)(19,25)(24,27), [128X[104X
    [4X[28X  ( 4, 5, 7,10)( 6, 9,12,16)( 8,11,14,18)(13,17,20,23)(15,19,22,25)[128X[104X
    [4X[28X    (21,24,26,27), ( 4, 6, 7,12)( 5, 9,10,16)( 8,13,14,20)(11,17,18,23)[128X[104X
    [4X[28X    (15,21,22,26)(19,24,25,27), ( 4, 7)( 5,10)( 6,12)( 8,14)( 9,16)(11,18)[128X[104X
    [4X[28X    (13,20)(15,22)(17,23)(19,25)(21,26)(24,27), ( 1, 2, 3), [128X[104X
    [4X[28X  ( 4, 8,15)( 5,11,19)( 6,13,21)( 7,14,22)( 9,17,24)(10,18,25)(12,20,26)[128X[104X
    [4X[28X    (16,23,27) ][128X[104X
    [4X[28X  [ (2,6)(3,5), (1,2,3,4,5,6), (1,3,5)(2,4,6) ][128X[104X
    [4X[28X: Source Homomorphism maps source generators to:[128X[104X
    [4X[28X  [ ( 6,12)( 8,15)( 9,16)(11,19)(13,26)(14,22)(17,27)(18,25)(20,21)(23,24), [128X[104X
    [4X[28X  ( 2, 3)( 5,10)( 9,16)(11,18)(17,23)(19,25)(24,27), [128X[104X
    [4X[28X  ( 4, 5, 7,10)( 6, 9,12,16)( 8,11,14,18)(13,17,20,23)(15,19,22,25)[128X[104X
    [4X[28X    (21,24,26,27), ( 4, 6, 7,12)( 5, 9,10,16)( 8,13,14,20)(11,17,18,23)[128X[104X
    [4X[28X    (15,21,22,26)(19,24,25,27), ( 4, 7)( 5,10)( 6,12)( 8,14)( 9,16)(11,18)[128X[104X
    [4X[28X    (13,20)(15,22)(17,23)(19,25)(21,26)(24,27), ( 1, 2, 3), [128X[104X
    [4X[28X  ( 4, 8,15)( 5,11,19)( 6,13,21)( 7,14,22)( 9,17,24)(10,18,25)(12,20,26)[128X[104X
    [4X[28X    (16,23,27) ][128X[104X
    [4X[28X: Range Homomorphism maps range generators to:[128X[104X
    [4X[28X  [ (2,6)(3,5), (1,2,3,4,5,6), (1,3,5)(2,4,6) ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X3.4 [33X[0;0YOperations on morphisms[133X[101X
  
  [1X3.4-1 CompositionMorphism[101X
  
  [33X[1;0Y[29X[2XCompositionMorphism[102X( [3Xmap2[103X, [3Xmap1[103X ) [32X operation[133X
  
  [33X[0;0YComposition  of  morphisms  (written  [10X(<map1> * <map2>)[110X when maps act on the
  right) calls the [2XCompositionMorphism[102X function for maps (acting on the left),
  applied to the appropriate type of 2d-mapping.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XH2 := Subgroup(G2,[G2.3,G2.4,G2.6,G2.7]);  SetName( H2, "H2" );[127X[104X
    [4X[28XGroup([ f3, f4, f6, f7 ])[128X[104X
    [4X[25Xgap>[125X [27Xc6 := Subgroup( d12, [b,c] );  SetName( c6, "c6" );[127X[104X
    [4X[28XGroup([ f2, f3 ])[128X[104X
    [4X[25Xgap>[125X [27XSC2 := Sub2DimensionalGroup( C2, H2, c6 );[127X[104X
    [4X[28X[H2=>c6][128X[104X
    [4X[25Xgap>[125X [27XIsCat1Group( SC2 );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xinc2 := InclusionMorphism2DimensionalDomains( C2, SC2 );[127X[104X
    [4X[28X[[H2=>c6] => [G2=>d12]][128X[104X
    [4X[25Xgap>[125X [27XCompositionMorphism( iso2, inc );                  [127X[104X
    [4X[28X[[H2=>c6] => P[G2=>d12]][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X3.4-2 Kernel[101X
  
  [33X[1;0Y[29X[2XKernel[102X( [3Xmap[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XKernel2DimensionalMapping[102X( [3Xmap[103X ) [32X attribute[133X
  
  [33X[0;0YThe  kernel  of  a morphism of crossed modules is a normal subcrossed module
  whose  groups  are  the  kernels of the source and target homomorphisms. The
  inclusion of the kernel is a standard example of a crossed square, but these
  have not yet been implemented.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xc2 := Group( (19,20) );                                    [127X[104X
    [4X[28XGroup([ (19,20) ])[128X[104X
    [4X[25Xgap>[125X [27XX0 := XModByNormalSubgroup( c2, c2 );  SetName( X0, "X0" );[127X[104X
    [4X[28X[Group( [ (19,20) ] )->Group( [ (19,20) ] )][128X[104X
    [4X[25Xgap>[125X [27X SX2 := Source( X2 );;[127X[104X
    [4X[25Xgap>[125X [27XgenSX2 := GeneratorsOfGroup( SX2 ); [127X[104X
    [4X[28X[ f1, f4, f5, f7 ][128X[104X
    [4X[25Xgap>[125X [27Xsigma0 := GroupHomomorphismByImages(SX2,c2,genSX2,[(19,20),(),(),()]);[127X[104X
    [4X[28X[ f1, f4, f5, f7 ] -> [ (19,20), (), (), () ][128X[104X
    [4X[25Xgap>[125X [27Xrho0 := GroupHomomorphismByImages(d12,c2,[a1,a2,a3],[(19,20),(),()]);[127X[104X
    [4X[28X[ f1, f2, f3 ] -> [ (19,20), (), () ][128X[104X
    [4X[25Xgap>[125X [27Xmor0 := XModMorphism( X2, X0, sigma0, rho0 );;           [127X[104X
    [4X[25Xgap>[125X [27XK0 := Kernel( mor0 );;[127X[104X
    [4X[25Xgap>[125X [27XStructureDescription( K0 );[127X[104X
    [4X[28X[ "C12", "C6" ][128X[104X
  [4X[32X[104X
  
