GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it

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<!--  gp2map.xml            XMod documentation            Chris Wensley  -->
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<!--                                                        & Murat Alp  -->
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<!--  Copyright (C) 2001-2017, Chris Wensley et al,                      --> 
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<!--  School of Computer Science, Bangor University, U.K.                --> 
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<!--                                                                     -->
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<?xml version="1.0" encoding="UTF-8"?>
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<Chapter Label="chap-gpmap2">
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<Heading>2d-mappings</Heading>
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<Index>2d-mapping</Index>
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<Section><Heading>Morphisms of 2-dimensional groups</Heading>
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<Index>morphism of 2d-group</Index>
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<Index>crossed module morphism</Index>
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This chapter describes morphisms of (pre-)crossed modules
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and (pre-)cat1-groups.
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<ManSection>
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   <Attr Name="Source"
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         Arg="map" />
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   <Attr Name="Range"
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         Arg="map" />
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   <Attr Name="SourceHom"
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         Arg="map" />
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   <Attr Name="RangeHom"
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         Arg="map" />
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<Description>
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Morphisms of <E>2-dimensional groups</E> are implemented as 
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<E>2-dimensional mappings</E>.
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These have a pair of 2-dimensional groups as source and range, 
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together with two group homomorphisms mapping between corresponding 
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source and range groups.
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These functions return <C>fail</C> when invalid data is supplied.
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</Description>
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</ManSection>
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</Section>
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<Section><Heading>Morphisms of pre-crossed modules</Heading>
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<Index>morphism</Index>
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<ManSection>
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   <Prop Name="IsXModMorphism"
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         Arg="map" />
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   <Prop Name="IsPreXModMorphism"
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         Arg="map" />
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<Description>
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A morphism between two pre-crossed modules 
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<M>\mathcal{X}_{1} = (\partial_1 : S_1 \to R_1)</M> and  
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<M>\mathcal{X}_{2} = (\partial_2 : S_2 \to R_2)</M> 
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is a pair  <M>(\sigma, \rho)</M>, where 
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<M>\sigma : S_1 \to S_2</M> and <M>\rho : R_1 \to R_2</M> 
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commute with the two boundary maps
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and are morphisms for the two actions:
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<Display>
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\partial_2 \circ \sigma ~=~ \rho \circ \partial_1, \qquad
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\sigma(s^r) ~=~ (\sigma s)^{\rho r}.
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</Display>
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Here is a diagram of the situation. 
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<Display><![CDATA[
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\vcenter{\xymatrix{
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  S_1 \ar[rr]^{\sigma} \ar[dd]_{\partial_1}
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     && S_2 \ar[dd]^{\partial_2} \\
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     &&  \\
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  R_1 \ar[rr]_{\rho}
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     && R_2 
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}}
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]]></Display>
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Here  <M>\sigma</M> is the <C>SourceHom</C>  
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and  <M>\rho</M>  is the <C>RangeHom</C>  of the morphism. 
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When <M>\mathcal{X}_{1} = \mathcal{X}_{2}</M>
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and <M>\sigma, \rho</M> are automorphisms then 
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<M>(\sigma, \rho)</M>  is an automorphism of <M>\mathcal{X}_1</M>. 
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The group of automorphisms is denoted 
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by <M>{\rm Aut}(\mathcal{X}_1 )</M>. 
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</Description>
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</ManSection>
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<ManSection>
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   <Prop Name="IsInjective"
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         Arg="map" />
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   <Prop Name="IsSurjective"
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         Arg="map" />
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   <Prop Name="IsSingleValued"
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         Arg="map" />
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   <Prop Name="IsTotal"
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         Arg="map" />
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   <Prop Name="IsBijective"
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         Arg="map" />
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   <Prop Name="IsEndo2DimensionalMapping"
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         Arg="map" />
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<Description>
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The usual properties of mappings are easily checked.
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It is usually sufficient to verify that both the <C>SourceHom</C> and the
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<C>RangeHom</C> have the required property.
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</Description>
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</ManSection>
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<ManSection>
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   <Func Name="XModMorphism"
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         Arg="args" />
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   <Oper Name="XModMorphismByHoms"
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         Arg="X1 X2 sigma rho" />
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   <Func Name="PreXModMorphism"
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         Arg="args" />
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   <Oper Name="PreXModMorphismByHoms"
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         Arg="P1 P2 sigma rho" />
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   <Oper Name="InclusionMorphism2DimensionalDomains"
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         Arg="X1 S1" />
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   <Oper Name="InnerAutomorphismXMod"
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         Arg="X1 r" />
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   <Attr Name="IdentityMapping"
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         Arg="X1" />
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<Description>
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These are the constructors for morphisms of pre-crossed and crossed modules.
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<P/>
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In the following example we construct a simple automorphism of 
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the crossed module  <C>X1</C>  constructed in the previous chapter.
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</Description>
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</ManSection>
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<P/>
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<Index>display a 2d-mapping</Index>
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<Index>order of a 2d-automorphism</Index>
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<Example>
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<![CDATA[
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gap> sigma1 := GroupHomomorphismByImages( c5, c5, [ (5,6,7,8,9) ]
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        [ (5,9,8,7,6) ] );;
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gap> rho1 := IdentityMapping( Range( X1 ) );
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IdentityMapping( PAut(c5) )
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gap> mor1 := XModMorphism( X1, X1, sigma1, rho1 );
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[[c5->PAut(c5))] => [c5->PAut(c5))]] 
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gap> Display( mor1 );
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Morphism of crossed modules :-
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: Source = [c5->PAut(c5))] with generating sets:
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  [ (5,6,7,8,9) ]
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  [ (1,2,3,4) ]
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: Range = Source
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: Source Homomorphism maps source generators to:
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  [ (5,9,8,7,6) ]
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: Range Homomorphism maps range generators to:
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  [ (1,2,3,4) ]
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gap> IsAutomorphism2DimensionalDomain( mor1 );
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true 
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gap> Order( mor1 );
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2
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gap> RepresentationsOfObject( mor1 );
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[ "IsComponentObjectRep", "IsAttributeStoringRep", "Is2DimensionalMappingRep" ]
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gap> KnownPropertiesOfObject( mor1 );
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[ "CanEasilyCompareElements", "CanEasilySortElements", "IsTotal", 
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  "IsSingleValued", "IsInjective", "IsSurjective", "RespectsMultiplication", 
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  "IsPreXModMorphism", "IsXModMorphism", "IsEndomorphism2DimensionalDomain", 
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  "IsAutomorphism2DimensionalDomain" ]
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gap> KnownAttributesOfObject( mor1 );
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[ "Name", "Order", "Range", "Source", "SourceHom", "RangeHom" ]
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]]>
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</Example>
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<ManSection>
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   <Attr Name="IsomorphismPerm2DimensionalGroup"
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         Arg="obj" />
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   <Attr Name="IsomorphismPc2DimensionalGroup"
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         Arg="obj" />
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   <Oper Name="IsomorphismByIsomorphisms"
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         Arg="D list" />
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<Description>
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When <M>{\mathcal D}</M> is a <M>2</M>-dimensional domain 
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with source <M>S</M> and range <M>R</M> 
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and <M>\sigma : S \to S',~ \rho : R \to R'</M> are isomorphisms, 
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then <C>IsomorphismByIsomorphisms(D,[sigma,rho])</C> returns an isomorphism 
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<M>(\sigma,\rho) : {\mathcal D} \to {\mathcal D}'</M> 
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where <M>{\mathcal D}'</M> has source <M>S'</M> and range <M>R'</M>.  
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Be sure to test <C>IsBijective</C> for the two functions 
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<M>\sigma,\rho</M> before applying this operation.  
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<P/> 
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Using <C>IsomorphismByIsomorphisms</C> with a pair of isomorphisms 
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obtained using <C>IsomorphismPermGroup</C> or <C>IsomorphismPcGroup</C>, 
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we may construct a crossed module or a cat1-group 
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of permutation groups or pc-groups.  
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</Description>
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</ManSection>
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<P/>
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<Example>
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<![CDATA[
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gap> q8 := SmallGroup(8,4);;   ## quaternion group 
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gap> Xq8 := XModByAutomorphismGroup( q8 );
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[Group( [ f1, f2, f3 ] )->Group( [ f1, f2, f3, f4 ] )]
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gap> iso := IsomorphismPerm2DimensionalGroup( Xq8 );;
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gap> Yq8 := Image( iso );
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[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) 
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 ] )->Group( [ (2,6,5,4), (1,2,4)(3,5,6), (2,5)(4,6), (1,3)(2,5) ] )]
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gap> s4 := SymmetricGroup(4);; 
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gap> isos4 := IsomorphismGroups( Range(Yq8), s4 );;
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gap> id := IdentityMapping( Source( Yq8 ) );; 
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gap> IsBijective( id );;  IsBijective( isos4 );;
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gap> mor := IsomorphismByIsomorphisms( Yq8, [id,isos4] );;
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gap> Zq8 := Image( mor );
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[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) 
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 ] )->SymmetricGroup( [ 1 .. 4 ] )]
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]]>
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</Example>
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</Section>
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<Section Label="sect-mor-pre-cat1">
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<Heading>Morphisms of pre-cat1-groups</Heading>
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A morphism of pre-cat1-groups from 
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<M>\mathcal{C}_1 = (e_1;t_1,h_1 : G_1 \to R_1)</M>
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to  <M>\mathcal{C}_2 = (e_2;t_2,h_2 : G_2 \to R_2)</M>  
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is a pair  <M>(\gamma, \rho)</M>  where
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<M>\gamma : G_1 \to G_2</M>  and  <M>\rho : R_1 \to R_2</M>  
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are homomorphisms satisfying
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<Display>
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h_2 \circ \gamma ~=~ \rho \circ h_1, \qquad 
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t_2 \circ \gamma ~=~ \rho \circ t_1, \qquad 
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e_2 \circ \rho ~=~ \gamma \circ e_1. 
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</Display>
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<ManSection>
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   <Prop Name="IsCat1Morphism"
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         Arg="map" />
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   <Prop Name="IsPreCat1Morphism"
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         Arg="map" />
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   <Func Name="Cat1Morphism"
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         Arg="args" />
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   <Oper Name="Cat1MorphismByHoms"
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         Arg="C1 C2 gamma rho" />
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   <Func Name="PreCat1Morphism"
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         Arg="args" />
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   <Oper Name="PreCat1MorphismByHoms"
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         Arg="P1 P2 gamma rho" />
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   <Oper Name="InclusionMorphism2DimensionalDomains"
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         Arg="C1 S1" />
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   <Oper Name="InnerAutomorphismCat1"
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         Arg="C1 r" />
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   <Attr Name="IdentityMapping"
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         Arg="C1" />
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<Description>
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For an example we form a second cat1-group <Code>C2=[g18=>s3a]</Code>, 
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similar to <C>C1</C> in <Ref Sect="mansect-cat1"/>, 
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then construct an isomorphism <M>(\gamma,\rho)</M> between them. 
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</Description>
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</ManSection> 
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<P/>
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<Example>
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<![CDATA[
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gap> t2 := GroupHomomorphismByImages(g18,s3a,g18gens,[(),(7,8,9),(8,9)]);;     
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gap> e2 := GroupHomomorphismByImages(s3a,g18,s3agens,[(4,5,6),(2,3)(5,6)]);;   
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gap> C2 := Cat1Group( t2, h1, e2 );; 
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gap> imgamma := [ (4,5,6), (1,2,3), (2,3)(5,6) ];; 
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gap> gamma := GroupHomomorphismByImages( g18, g18, g18gens, imgamma );;
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gap> rho := IdentityMapping( s3a );; 
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gap> mor := Cat1Morphism( C1, C2, gamma, rho );;
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gap> Display( mor );;
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Morphism of cat1-groups :- 
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: Source = [g18=>s3a] with generating sets:
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  [ (1,2,3), (4,5,6), (2,3)(5,6) ]
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  [ (7,8,9), (8,9) ]
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:  Range = [g18=>s3a] with generating sets:
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  [ (1,2,3), (4,5,6), (2,3)(5,6) ]
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  [ (7,8,9), (8,9) ]
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: Source Homomorphism maps source generators to:
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  [ (4,5,6), (1,2,3), (2,3)(5,6) ]
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: Range Homomorphism maps range generators to:
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  [ (7,8,9), (8,9) ]
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]]>
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</Example>
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<ManSection>
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   <Func Name="IsomorphismPermObject" 
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         Arg="obj" />
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   <Attr Name="IsomorphismPerm2DimensionalGroup" 
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         Arg="2DimensionalGroup" />
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   <Attr Name="IsomorphismFp2DimensionalGroup"
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         Arg="2DimensionalGroup" />
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   <Attr Name="IsomorphismPc2DimensionalGroup"
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         Arg="2DimensionalGroup" />
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   <Func Name="SmallerDegreePerm2DimensionalDomain"
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         Arg="2DimensionalDomain" />
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<Description>
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The global function <C>IsomorphismPermObject</C> 
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calls <C>IsomorphismPerm2DimensionalGroup</C>, 
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which constructs a morphism whose <C>SourceHom</C> and <C>RangeHom</C> 
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are calculated using <C>IsomorphismPermGroup</C> on the source and range.  
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Similarly <C>SmallerDegreePermutationRepresentation</C> may be used on the
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two groups to obtain <C>SmallerDegreePerm2DimensionalDomain</C>.
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Names are assigned automatically.
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</Description>
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</ManSection>
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<P/>
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<Example>
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<![CDATA[
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gap> iso2 := IsomorphismPerm2DimensionalGroup( C2 );
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[[G2=>d12] => [..]]
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gap> Display( iso2 );
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Morphism of cat1-groups :- 
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: Source = [G2=>d12] with generating sets:
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  [ f1, f2, f3, f4, f5, f6, f7 ]
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  [ f1, f2, f3 ]
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:  Range = P[G2=>d12] with generating sets:
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  [ ( 6,12)( 8,15)( 9,16)(11,19)(13,26)(14,22)(17,27)(18,25)(20,21)(23,24), 
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  ( 2, 3)( 5,10)( 9,16)(11,18)(17,23)(19,25)(24,27), 
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  ( 4, 5, 7,10)( 6, 9,12,16)( 8,11,14,18)(13,17,20,23)(15,19,22,25)
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    (21,24,26,27), ( 4, 6, 7,12)( 5, 9,10,16)( 8,13,14,20)(11,17,18,23)
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    (15,21,22,26)(19,24,25,27), ( 4, 7)( 5,10)( 6,12)( 8,14)( 9,16)(11,18)
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    (13,20)(15,22)(17,23)(19,25)(21,26)(24,27), ( 1, 2, 3), 
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  ( 4, 8,15)( 5,11,19)( 6,13,21)( 7,14,22)( 9,17,24)(10,18,25)(12,20,26)
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    (16,23,27) ]
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  [ (2,6)(3,5), (1,2,3,4,5,6), (1,3,5)(2,4,6) ]
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: Source Homomorphism maps source generators to:
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  [ ( 6,12)( 8,15)( 9,16)(11,19)(13,26)(14,22)(17,27)(18,25)(20,21)(23,24), 
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  ( 2, 3)( 5,10)( 9,16)(11,18)(17,23)(19,25)(24,27), 
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  ( 4, 5, 7,10)( 6, 9,12,16)( 8,11,14,18)(13,17,20,23)(15,19,22,25)
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    (21,24,26,27), ( 4, 6, 7,12)( 5, 9,10,16)( 8,13,14,20)(11,17,18,23)
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    (15,21,22,26)(19,24,25,27), ( 4, 7)( 5,10)( 6,12)( 8,14)( 9,16)(11,18)
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    (13,20)(15,22)(17,23)(19,25)(21,26)(24,27), ( 1, 2, 3), 
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  ( 4, 8,15)( 5,11,19)( 6,13,21)( 7,14,22)( 9,17,24)(10,18,25)(12,20,26)
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    (16,23,27) ]
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: Range Homomorphism maps range generators to:
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  [ (2,6)(3,5), (1,2,3,4,5,6), (1,3,5)(2,4,6) ]
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]]>
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</Example>
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</Section>
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<Section Label="sect-oper-mor">
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<Heading>Operations on morphisms</Heading>
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<Index>operations on morphisms</Index>
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<ManSection>
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   <Oper Name="CompositionMorphism"
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         Arg="map2 map1" />
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<Description>
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Composition of morphisms (written <C>(&lt;map1&gt; * &lt;map2&gt;)</C> 
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when maps act on the right) 
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calls the <C>CompositionMorphism</C> function for maps (acting on the left),
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applied to the appropriate type of 2d-mapping.
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</Description>
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</ManSection>
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<P/>
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<Example>
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<![CDATA[
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gap> H2 := Subgroup(G2,[G2.3,G2.4,G2.6,G2.7]);  SetName( H2, "H2" );
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Group([ f3, f4, f6, f7 ])
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gap> c6 := Subgroup( d12, [b,c] );  SetName( c6, "c6" );
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Group([ f2, f3 ])
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gap> SC2 := Sub2DimensionalGroup( C2, H2, c6 );
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[H2=>c6]
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gap> IsCat1Group( SC2 );
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true
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gap> inc2 := InclusionMorphism2DimensionalDomains( C2, SC2 );
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[[H2=>c6] => [G2=>d12]]
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gap> CompositionMorphism( iso2, inc );                  
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[[H2=>c6] => P[G2=>d12]]
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]]>
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</Example>
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<ManSection>
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   <Oper Name="Kernel"
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         Arg="map" />
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   <Attr Name="Kernel2DimensionalMapping"
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         Arg="map" />
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<Description>
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The kernel of a morphism of crossed modules is a normal subcrossed module 
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whose groups are the kernels of the source and target homomorphisms. 
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The inclusion of the kernel is a standard example of a crossed square, 
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but these have not yet been implemented.
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</Description>
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</ManSection>
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<P/>
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<Example>
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<![CDATA[
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gap> c2 := Group( (19,20) );                                    
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Group([ (19,20) ])
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gap> X0 := XModByNormalSubgroup( c2, c2 );  SetName( X0, "X0" );
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[Group( [ (19,20) ] )->Group( [ (19,20) ] )]
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gap>  SX2 := Source( X2 );;
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gap> genSX2 := GeneratorsOfGroup( SX2 ); 
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[ f1, f4, f5, f7 ]
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gap> sigma0 := GroupHomomorphismByImages(SX2,c2,genSX2,[(19,20),(),(),()]);
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[ f1, f4, f5, f7 ] -> [ (19,20), (), (), () ]
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gap> rho0 := GroupHomomorphismByImages(d12,c2,[a1,a2,a3],[(19,20),(),()]);
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[ f1, f2, f3 ] -> [ (19,20), (), () ]
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gap> mor0 := XModMorphism( X2, X0, sigma0, rho0 );;           
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gap> K0 := Kernel( mor0 );;
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gap> StructureDescription( K0 );
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[ "C12", "C6" ]
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]]></Example>
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</Section>
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</Chapter>
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