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class="nocss"> </span><a href="chap2_mj.html#X7CE4F14585F6D47 </div></div> <div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html Properties of cat<span class="SimpleMath">\(^1\)</span>-groups and pre-cat<span class="Simp </span></a> </span> <div class="ContSSBlock"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap2_mj.html#X78E03FAB84A57D0 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap2_mj.html#X7B7CF88F83B0129 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap2_mj.html#X82F10A59867C765 </div></div> <div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html </span> <div class="ContSSBlock"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap2_mj.html#X7BDEBBF17CE6A6D <span class="ContSS"><br /><span class="nocss"> </span><a href="chap2_mj.html#X7FBFC8C87FC1AC5 <span class="ContSS"><br /><span class="nocss"> </span><a 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href="chap2_mj.html#X8578AB6D7C1FC4F </div></div> </div> <h3>2 <span class="Heading">2d-groups : crossed modules and cat<span class="SimpleMath">\(^1\)< <p>The term <em>2d-group</em> refers to a set of equivalent categories of which the most common are th <p><a id="X7BAD9A7F7AFEEC89" name="X7BAD9A7F7AFEEC89"></a></p> <h4>2.1 <span class="Heading">Constructions for crossed modules</span></h4> <p>A crossed module (of groups) <span class="SimpleMath">\(\calX = (\partial : S \to R )\)</span> co <p class="center">\[ {\bf XMod\ 1} : \partial(s^r) = r^{-1} (\partial s) r = (\partial s)^r, \qquad {\bf XMod\ 2} : s_1^{\partial s_2} = s_2^{-1}s_1 s_2 = {s_1}^{s_2}. \]</p> <p>When only the first of these axioms is satisfied, the resulting structure is a <em>pre-crossed m <p>(Much of the literature on crossed modules uses left actions, but we have chosen to use right ac <p><a id="X7C8175AE7F76B586" name="X7C8175AE7F76B586"></a></p> <h5>2.1-1 XMod</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ XM <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ XM <p>The global function <code class="code">XMod</code> calls one of the standard constructions des <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">c5 := Group( (5,6,7,8,9) );;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">SetName( c5, "c5" );</span> <span class="GAPprompt">gap></span> <span class="GAPinput">id5 := IdentityMapping( c5 );;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">ac5 := AutomorphismGroup( c5 );; </sp <span class="GAPprompt">gap></span> <span class="GAPinput">act := MappingToOne( c5, ac5 );;</spa <span class="GAPprompt">gap></span> <span class="GAPinput">XMod( id5, act ) = XModByBoundaryAnd true </pre></div> <p><a id="X83050ED686776933" name="X83050ED686776933"></a></p> <h5>2.1-2 XModByNormalSubgroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ XM <p>A <em>conjugation crossed module</em> is the inclusion of a normal subgroup <span class="SimpleM <p><a id="X867B2D53832EF05E" name="X867B2D53832EF05E"></a></p> <h5>2.1-3 XModByTrivialAction</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ XM <p>A <em>trivial action crossed module</em> <span class="SimpleMath">\((\partial : S \to R)\)</span> <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">q8 := QuaternionGroup( IsPermGroup Group([ (1,5,3,7)(2,8,4,6), (1,2,3,4)(5,6,7,8) ]) <span class="GAPprompt">gap></span> <span class="GAPinput">SetName( q8, "q8" );</span> <span class="GAPprompt">gap></span> <span class="GAPinput">c2 := Centre( q8 ); </span> Group([ (1,3)(2,4)(5,7)(6,8) ]) <span class="GAPprompt">gap></span> <span class="GAPinput">SetName( c2, "<-1>" );</span> <span class="GAPprompt">gap></span> <span class="GAPinput">bdy := InclusionMappingGroups( q8, <span class="GAPprompt">gap></span> <span class="GAPinput">X8a := XModByTrivialAction( bdy );</ [<-1>->q8] <span class="GAPprompt">gap></span> <span class="GAPinput">c4 := Subgroup( q8, [q8.1] );;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">SetName( c4, "<i>" );</span> <span class="GAPprompt">gap></span> <span class="GAPinput">X8b := XModByNormalSubgroup( q8, c4 [<i>->q8] <span class="GAPprompt">gap></span> <span class="GAPinput">Display(X8b); </span> Crossed module [<i>->q8] :- : Source group has generators: [ (1,5,3,7)(2,8,4,6) ] : Range group q8 has generators: [ (1,5,3,7)(2,8,4,6), (1,2,3,4)(5,6,7,8) ] : Boundary homomorphism maps source generators to: [ (1,5,3,7)(2,8,4,6) ] : Action homomorphism maps range generators to automorphisms: (1,5,3,7)(2,8,4,6) --> { source gens --> [ (1,5,3,7)(2,8,4,6) ] } (1,2,3,4)(5,6,7,8) --> { source gens --> [ (1,7,3,5)(2,6,4,8) ] } These 2 automorphisms generate the group of automorphisms. </pre></div> <p><a id="X78B14FDA817CCEEF" name="X78B14FDA817CCEEF"></a></p> <h5>2.1-4 XModByAutomorphismGroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ XM <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ XM <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ XM <p>An <em>automorphism crossed module</em> has as range a subgroup <span class="SimpleMath">\(R\)</ <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">X5 := XModByAutomorphismGroup( c5 ) [c5 -> Aut(c5)] <span class="GAPprompt">gap></span> <span class="GAPinput">Display( X5 );</span> Crossed module [c5->Aut(c5)] :- : Source group c5 has generators: [ (5,6,7,8,9) ] : Range group Aut(c5) has generators: [ GroupHomomorphismByImages( c5, c5, [ (5,6,7,8,9) ], [ (5,7,9,6,8) ] ) ] : Boundary homomorphism maps source generators to: [ IdentityMapping( c5 ) ] : Action homomorphism maps range generators to automorphisms: GroupHomomorphismByImages( c5, c5, [ (5,6,7,8,9) ], [ (5,7,9,6,8) ] ) --> { source gens --> [ (5,7,9,6,8) ] } This automorphism generates the group of automorphisms. </pre></div> <p><a id="X7D0F6FAA7AF69844" name="X7D0F6FAA7AF69844"></a></p> <h5>2.1-5 XModByCentralExtension</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ XM <p>A <em>central extension crossed module</em> has as boundary a surjection <span class="SimpleMat <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">gen12 := [ (1,2,3,4,5,6), (2,6)(3,5 <span class="GAPprompt">gap></span> <span class="GAPinput">d12 := Group( gen12 );; </span> <span class="GAPprompt">gap></span> <span class="GAPinput">gen6 := [ (7,8,9), (8,9) ];;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">s3 := Group( gen6 );;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">SetName( d12, "d12" ); SetName( s3, " <span class="GAPprompt">gap></span> <span class="GAPinput">pr12 := GroupHomomorphismByImages <span class="GAPprompt">gap></span> <span class="GAPinput">Kernel( pr12 ) = Centre( d12 );</span> true <span class="GAPprompt">gap></span> <span class="GAPinput">X12 := XModByCentralExtension( pr1 <span class="GAPprompt">gap></span> <span class="GAPinput">Display( X12 ); </span> Crossed module [d12->s3] :- : Source group d12 has generators: [ (1,2,3,4,5,6), (2,6)(3,5) ] : Range group s3 has generators: [ (7,8,9), (8,9) ] : Boundary homomorphism maps source generators to: [ (7,8,9), (8,9) ] : Action homomorphism maps range generators to automorphisms: (7,8,9) --> { source gens --> [ (1,2,3,4,5,6), (1,3)(4,6) ] } (8,9) --> { source gens --> [ (1,6,5,4,3,2), (2,6)(3,5) ] } These 2 automorphisms generate the group of automorphisms. </pre></div> <p><a id="X84FA2B0A795B6997" name="X84FA2B0A795B6997"></a></p> <h5>2.1-6 XModByPullback</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ XM <p>Let <span class="SimpleMath">\(\calX_0 = (\mu : M \to P)\)</span> be a crossed module. If <span cla <p>The example forms a pullback of the crossed module <code class="code">X12</code> of the previous <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">gens4 := [ (11,12), (12,13), (13,14) <span class="GAPprompt">gap></span> <span class="GAPinput">s4 := Group( gens4 );; </span> <span class="GAPprompt">gap></span> <span class="GAPinput">theta := GroupHomomorphismByImage <span class="GAPprompt">gap></span> <span class="GAPinput">X1 := XModByPullback( X12, theta );; < <span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription( Source( X1 ) "C2 x S4" <span class="GAPprompt">gap></span> <span class="GAPinput">SetName( s4, "s4" ); SetName( Source <span class="GAPprompt">gap></span> <span class="GAPinput">infoX1 := PullbackInfo( Source( X1 ) <span class="GAPprompt">gap></span> <span class="GAPinput">infoX1!.directProduct;</span> Group([ (1,2,3,4,5,6), (2,6)(3,5), (7,8), (8,9), (9,10) ]) <span class="GAPprompt">gap></span> <span class="GAPinput">infoX1!.projections[1];</span> [ (7,8)(9,10), (7,9)(8,10), (2,6)(3,5)(8,9), (1,5,3)(2,6,4)(8,10,9), (1,6,5,4,3,2)(8,9,10) ] -> [ (), (), (2,6)(3,5), (1,5,3)(2,6,4), (1,6,5,4,3,2) ] <span class="GAPprompt">gap></span> <span class="GAPinput">infoX1!.projections[2];</span> [ (7,8)(9,10), (7,9)(8,10), (2,6)(3,5)(8,9), (1,5,3)(2,6,4)(8,10,9), (1,6,5,4,3,2)(8,9,10) ] -> [ (11,12)(13,14), (11,13)(12,14), (12,13), (12,14,13), (12,13,14) ] </pre></div> <p><a id="X824631577864961E" name="X824631577864961E"></a></p> <h5>2.1-7 XModByAbelianModule</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ XM <p>A <em>crossed abelian module</em> has an abelian module as source and the zero map as boundary. See <p><a id="X81704DFB795C0D29" name="X81704DFB795C0D29"></a></p> <h5>2.1-8 DirectProduct</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Di <p>The direct product <span class="SimpleMath">\(\calX_{1} \times \calX_{2}\)</span> of two cro <p>The example constructs the product of the two crossed modules formed in subsection <code class <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">X8ab := DirectProduct( X8a, X8b );</s [[<-1>->q8]x[<i>->q8]] <span class="GAPprompt">gap></span> <span class="GAPinput">infoX8ab := DirectProductInfo( X8a rec( embeddings := [ [[<-1>->q8] => [<-1>x<i>->q8xq8]], [[<i>->q8] => [<-1>x<i>->q8xq8]] ], objects := [ [<-1>->q8], [<i>->q8] ] , projections := [ [[<-1>x<i>->q8xq8] => [<-1>->q8]], [[<-1>x<i>->q8xq8] => [<i>->q8]] ] ) <span class="GAPprompt">gap></span> <span class="GAPinput">DirectProduct( X8a, X8b, X12 );</spa [[[<-1>->q8]x[<i>->q8]]x[d12->s3]] </pre></div> <p><a id="X790248A67CB9C33A" name="X790248A67CB9C33A"></a></p> <h5>2.1-9 Source</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ So <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ra <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Bo <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ XM <p>The following attributes are used in the construction of a crossed module <code class="code">X <ul> <li><p><code class="code">Source(X0)</code> and <code class="code">Range(X0)</code> are the source <s </li> <li><p><code class="code">XModAction(X0)</code> is a homomorphism from <span class="SimpleMath">\ </li> </ul> <p>(Up until version 2.63 there was an additional attribute <code class="code">AutoGroup</code>, <p>The example uses the crossed module <code class="code">X12</code> constructed in subsection <co <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">[ Source( X12 ), Range( X12 ) ]; </span> [ d12, s3 ] <span class="GAPprompt">gap></span> <span class="GAPinput">Boundary( X12 ); </span> [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (7,8,9), (8,9) ] <span class="GAPprompt">gap></span> <span class="GAPinput">XModAction( X12 );</span> [ (7,8,9), (8,9) ] -> [ [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,2,3,4,5,6), (1,3)(4,6) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,6,5,4,3,2), (2,6)(3,5) ] ] </pre></div> <p><a id="X7AF6602C87845F1D" name="X7AF6602C87845F1D"></a></p> <h5>2.1-10 ImageElmXModAction</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Im <p>This function returns the element <span class="SimpleMath">\(s^r\)</span> given by <code class <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">ImageElmXModAction( X12, (1,2,3,4 (1,6,5,4,3,2) </pre></div> <p><a id="X7846A7D37957B89E" name="X7846A7D37957B89E"></a></p> <h5>2.1-11 Size2d</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Si <p>The standard operation <code class="code">Size</code> cannot be used for crossed modules becau <p>In the simple example below, <code class="code">X5</code> is the automorphism crossed module co <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">Size2d( X5 ); </span> [ 5, 4 ] </pre></div> <p><a id="X85516B19803C01C0" name="X85516B19803C01C0"></a></p> <h5>2.1-12 Name</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Na <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Id <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ex <p>More familiar attributes are <code class="code">Name</code> and <code class="code">IdGroup</co <p>The <code class="func">ExternalSetXMod</code> for a crossed module is the source group conside <p>The <code class="code">Display</code> function is used to print details of 2d-groups.</p> <p>The <code class="code">Print</code> statements at the end of the example list the <strong class=" <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">IdGroup( X5 ); </span> [ [ 5, 1 ], [ 4, 1 ] ] <span class="GAPprompt">gap></span> <span class="GAPinput">ext := ExternalSetXMod( X5 ); </span> <xset:[ (), (5,6,7,8,9), (5,7,9,6,8), (5,8,6,9,7), (5,9,8,7,6) ]> <span class="GAPprompt">gap></span> <span class="GAPinput">Orbits( ext );</span> [ [ () ], [ (5,6,7,8,9), (5,7,9,6,8), (5,9,8,7,6), (5,8,6,9,7) ] ] <span class="GAPprompt">gap></span> <span class="GAPinput">a := GeneratorsOfGroup( Range( X5 ) ) [ (5,6,7,8,9) ] -> [ (5,9,8,7,6) ] <span class="GAPprompt">gap></span> <span class="GAPinput">ImageElmXModAction( X5, (5,7,9,6, (5,8,6,9,7) <span class="GAPprompt">gap></span> <span class="GAPinput">Print( RepresentationsOfObject(X [ "IsComponentObjectRep", "IsAttributeStoringRep", "IsPreXModObj" ] <span class="GAPprompt">gap></span> <span class="GAPinput">Print( KnownAttributesOfObject(X [ "Name", "Range", "Source", "IdGroup", "Boundary", "Size2d", "XModAction", "ExternalSetXMod", "HigherDimension" ] </pre></div> <p><a id="X7CF622538749FE73" name="X7CF622538749FE73"></a></p> <h4>2.2 <span class="Heading">Properties of crossed modules</span></h4> <p>The underlying category structures for the objects constructed in this chapter follow the se <p>There are then a variety of properties associated with crossed modules, starting with <code cl <p><a id="X7E77E6B881B1CE50" name="X7E77E6B881B1CE50"></a></p> <h5>2.2-1 IsXMod</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>A structure which has <code class="code">Is2DimensionalGroup</code> is a precrossed module or <p>There are also properties corresponding to the various construction methods which are liste <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">[ IsTrivial( X5 ), IsNonTrivial( X5 ) [ false, true, true ] <span class="GAPprompt">gap></span> <span class="GAPinput">kpoX5 := KnownPropertiesOfObject( <span class="GAPprompt">gap></span> <span class="GAPinput">ForAll( [ "IsTrivial", "IsNonTrivi <span class="GAPprompt">></span> <span class="GAPinput"> "CanEasilyCompareElements", "CanEas <span class="GAPprompt">></span> <span class="GAPinput"> "IsGeneratorsOfSemigroup", "IsPreXM <span class="GAPprompt">></span> <span class="GAPinput"> "IsAutomorphismGroup2DimensionalGr <span class="GAPprompt">></span> <span class="GAPinput"> s -> s in kpoX5 ); </span> true <span class="GAPprompt">gap></span> <span class="GAPinput">Is2DimensionalGroup(X5);</span> true <span class="GAPprompt">gap></span> <span class="GAPinput">IsAutomorphismGroup2Dimensional true </pre></div> <p><a id="X7884284383284A87" name="X7884284383284A87"></a></p> <h5>2.2-2 SubXMod</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Su <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Tr <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ No <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>With the standard crossed module constructors listed above as building blocks, sub-crossed <ul> <li><p><span class="SimpleMath">\(N,M\)</span> are normal subgroups of <span class="SimpleMath">\ </li> <li><p><span class="SimpleMath">\(\delta\)</span> is the restriction of <span class="SimpleMath"> </li> <li><p><span class="SimpleMath">\(n^r \in N\)</span> for all <span class="SimpleMath">\(n \in N,~r \ </li> <li><p><span class="SimpleMath">\((s^{-1})^ms \in N\)</span> for all <span class="SimpleMath">\(m </li> </ul> <p>These conditions ensure that <span class="SimpleMath">\(M \ltimes N\)</span> is normal in the s <p>A method for <code class="code">IsNormal</code> for precrossed modules is provided. See sectio <p>The five normal subcrossed modules of <code class="code">X4</code> found in the following examp <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">s4 := Group( (1,2), (2,3), (3,4) );; </ <span class="GAPprompt">gap></span> <span class="GAPinput">a4 := Subgroup( s4, [ (1,2,3), (2,3,4 <span class="GAPprompt">gap></span> <span class="GAPinput">k4 := Subgroup( a4, [ (1,2)(3,4), (1, <span class="GAPprompt">gap></span> <span class="GAPinput">SetName(s4,"s4"); SetName(a4,"a4 <span class="GAPprompt">gap></span> <span class="GAPinput">X4 := XModByNormalSubgroup( s4, a4 ) [a4->s4] <span class="GAPprompt">gap></span> <span class="GAPinput">Y4 := SubXMod( X4, k4, a4 ); </span> [k4->a4] <span class="GAPprompt">gap></span> <span class="GAPinput">IsNormal( X4, Y4 );</span> true <span class="GAPprompt">gap></span> <span class="GAPinput">NX4 := NormalSubXMods( X4 );;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">Length( NX4 ); </span> 5 </pre></div> <p><a id="X7D8165F77B23BCF6" name="X7D8165F77B23BCF6"></a></p> <h5>2.2-3 KernelCokernelXMod</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ke <p>Let <span class="SimpleMath">\(\calX = (\partial : S \to R)\)</span>. If <span class="SimpleMat <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">d8d8 := Group( (1,2,3,4), (1,3), (5, <span class="GAPprompt">gap></span> <span class="GAPinput">X88 := XModByAutomorphismGroup( d8 <span class="GAPprompt">gap></span> <span class="GAPinput">Size2d( X88 );</span> [ 64, 2048 ] <span class="GAPprompt">gap></span> <span class="GAPinput">Y88 := KernelCokernelXMod( X88 );;</ <span class="GAPprompt">gap></span> <span class="GAPinput">IdGroup(Y88);</span> [ [ 4, 2 ], [ 128, 928 ] ] <span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription( Y88 );</span> [ "C2 x C2", "(D8 x D8) : C2" ] </pre></div> <p><a id="X7D435B6279032D4D" name="X7D435B6279032D4D"></a></p> <h4>2.3 <span class="Heading">Pre-crossed modules</span></h4> <p><a id="X8487BE427858C5C9" name="X8487BE427858C5C9"></a></p> <h5>2.3-1 PreXModByBoundaryAndAction</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Pr <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Pr <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Su <p>If axiom <span class="SimpleMath">\({\bf XMod\ 2}\)</span> is <em>not</em> satisfied, the corresp <p>A special case of this operation is when the range is a trivial group (not necessarily a subgrou <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">b1 := (11,12,13,14,15,16,17,18);; <span class="GAPprompt">gap></span> <span class="GAPinput">d16 := Group( b1, b2 );;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">sk4 := Subgroup( d16, [ b1^4, b2 ] );;</ <span class="GAPprompt">gap></span> <span class="GAPinput">SetName( d16, "d16" ); SetName( sk4, <span class="GAPprompt">gap></span> <span class="GAPinput">bdy16 := GroupHomomorphismByImage <span class="GAPprompt">gap></span> <span class="GAPinput">aut1 := GroupHomomorphismByImages <span class="GAPprompt">gap></span> <span class="GAPinput">aut2 := GroupHomomorphismByImages <span class="GAPprompt">gap></span> <span class="GAPinput">aut16 := Group( [ aut1, aut2 ] );;</spa <span class="GAPprompt">gap></span> <span class="GAPinput">act16 := GroupHomomorphismByImage <span class="GAPprompt">gap></span> <span class="GAPinput">P16 := PreXModByBoundaryAndAction [d16->sk4] <span class="GAPprompt">gap></span> <span class="GAPinput">IsXMod( P16 );</span> false <span class="GAPprompt">gap></span> <span class="GAPinput">Q16 := PreXModWithTrivialRange( d1 [d16->Group( [ () ] )] <span class="GAPprompt">gap></span> <span class="GAPinput">SQ16 := SubPreXMod( Q16, sk4, Group( <span class="GAPprompt">gap></span> <span class="GAPinput">Display(SQ16);</span> Crossed module :- : Source group has generators: [ (11,15)(12,16)(13,17)(14,18), (12,18)(13,17)(14,16) ] : Range group has generators: [ () ] : Boundary homomorphism maps source generators to: [ (), () ] The automorphism group is trivial </pre></div> <p><a id="X8527F4C07A8F359E" name="X8527F4C07A8F359E"></a></p> <h5>2.3-2 PeifferSubgroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Pe <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ XM <p>The <em>Peiffer subgroup</em> <span class="SimpleMath">\(P\)</span> of a pre-crossed module <span <p class="center">\[ \lfloor s_1,s_2 \rfloor ~=~ (s_1^{-1})^{\partial s_2}~s_2^{-1}~s_1~s_2 ~=~ \langle \partial s_2, s_1 \rangle\ [s_1,s_2]~. \]</p> <p>Then <span class="SimpleMath">\(\calP = (0 : P \to \{1_R\})\)</span> is a normal sub-pre-crosse <p>In the following example the Peiffer subgroup is cyclic of size <span class="SimpleMath">\(4\ <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">P := PeifferSubgroup( P16 );</span> Group( [ (11,15)(12,16)(13,17)(14,18), (11,17,15,13)(12,18,16,14) ] ) <span class="GAPprompt">gap></span> <span class="GAPinput">X16 := XModByPeifferQuotient( P16 ) Peiffer([d16->sk4]) <span class="GAPprompt">gap></span> <span class="GAPinput">Display( X16 );</span> Crossed module Peiffer([d16->sk4]) :- : Source group has generators: [ f1, f2 ] : Range group has generators: [ (11,15)(12,16)(13,17)(14,18), (12,18)(13,17)(14,16) ] : Boundary homomorphism maps source generators to: [ (12,18)(13,17)(14,16), (11,15)(12,16)(13,17)(14,18) ] The automorphism group is trivial <span class="GAPprompt">gap></span> <span class="GAPinput">iso16 := IsomorphismPermGroup( Sou <span class="GAPprompt">gap></span> <span class="GAPinput">S16 := Image( iso16 );</span> Group([ (1,2), (3,4) ]) </pre></div> <p><a id="X7AAABC1D7E110988" name="X7AAABC1D7E110988"></a></p> <h4>2.4 <span class="Heading">Cat<span class="SimpleMath">\(^1\)</span>-groups and pre-cat<spa <p>In <a href="chapBib_mj.html#biBL1">[Lod82]</a>, Loday reformulated the notion of a crossed mo <p class="center">\[ {\bf Cat\ 1} : ~t \circ e ~=~ h \circ e = {\rm id}_R, \qquad {\bf Cat\ 2} : ~[\ker t, \ker h] ~=~ \{ 1_G \}. \]</p> <p>It follows that <span class="SimpleMath">\(\;t \circ e \circ h = h,~ h \circ e \circ t = t,~ t \circ <p>See section <code class="func">IsPreCat1GroupWithIdentityEmbedding</code> (<a href="chap2 <p><a id="X7CF4C37F87D27EBA" name="X7CF4C37F87D27EBA"></a></p> <h5>2.4-1 Cat1Group</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ca <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Pr <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Pr <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Pr <p>The global functions <code class="code">Cat1Group</code> and <code class="code">PreCat1Group< <ul> <li><p>as <code class="code">Cat1Group(t,h,e);</code> when <span class="SimpleMath">\(t,h,e\)</s </li> <li><p>as <code class="code">Cat1Group(t,h);</code> when <span class="SimpleMath">\(t,h\)</span> a </li> <li><p>as <code class="code">Cat1Group(t);</code> when <span class="SimpleMath">\(t=h\)</span> is a </li> <li><p>as <code class="code">Cat1Group(t,e);</code> when <span class="SimpleMath">\(t=h\)</span> a </li> <li><p>as <code class="code">Cat1Group(i,j,k);</code> when <span class="SimpleMath">\(i,j,k\)</s </li> </ul> <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">g18gens := [ (1,2,3), (4,5,6), (2,3) <span class="GAPprompt">gap></span> <span class="GAPinput">s3agens := [ (7,8,9), (8,9) ];; </span> <span class="GAPprompt">gap></span> <span class="GAPinput">g18 := Group( g18gens );; SetName( g1 <span class="GAPprompt">gap></span> <span class="GAPinput">s3a := Group( s3agens );; SetName( s3 <span class="GAPprompt">gap></span> <span class="GAPinput">t1 := GroupHomomorphismByImages(g [ (1,2,3), (4,5,6), (2,3)(5,6) ] -> [ (7,8,9), (), (8,9) ] <span class="GAPprompt">gap></span> <span class="GAPinput">h1 := GroupHomomorphismByImages(g [ (1,2,3), (4,5,6), (2,3)(5,6) ] -> [ (7,8,9), (7,8,9), (8,9) ] <span class="GAPprompt">gap></span> <span class="GAPinput">e1 := GroupHomomorphismByImages(s [ (7,8,9), (8,9) ] -> [ (1,2,3), (2,3)(5,6) ] <span class="GAPprompt">gap></span> <span class="GAPinput">C18 := Cat1Group( t1, h1, e1 );</span> [g18=>s3a] </pre></div> <p><a id="X7C4FFC4086531157" name="X7C4FFC4086531157"></a></p> <h5>2.4-2 Source</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ So <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ra <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ta <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ He <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ra <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ke <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Bo <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Na <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Si <p>These are the attributes of a cat<span class="SimpleMath">\(^1\)</span>-group <span class="Si <p>The maps <span class="SimpleMath">\(t,h\)</span> are often referred to as the <em>source</em> and <e <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">[ Source( C18 ), Range( C18 ) ];</span> [ g18, s3a ] <span class="GAPprompt">gap></span> <span class="GAPinput">TailMap( C18 );</span> [ (1,2,3), (4,5,6), (2,3)(5,6) ] -> [ (7,8,9), (), (8,9) ] <span class="GAPprompt">gap></span> <span class="GAPinput">HeadMap( C18 );</span> [ (1,2,3), (4,5,6), (2,3)(5,6) ] -> [ (7,8,9), (7,8,9), (8,9) ] <span class="GAPprompt">gap></span> <span class="GAPinput">RangeEmbedding( C18 );</span> [ (7,8,9), (8,9) ] -> [ (1,2,3), (2,3)(5,6) ] <span class="GAPprompt">gap></span> <span class="GAPinput">Kernel( C18 );</span> Group([ (4,5,6) ]) <span class="GAPprompt">gap></span> <span class="GAPinput">KernelEmbedding( C18 );</span> [ (4,5,6) ] -> [ (4,5,6) ] <span class="GAPprompt">gap></span> <span class="GAPinput">Name( C18 );</span> "[g18=>s3a]" <span class="GAPprompt">gap></span> <span class="GAPinput">Size2d( C18 );</span> [ 18, 6 ] <span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription( C18 );</span> [ "(C3 x C3) : C2", "S3" ] </pre></div> <p>The next four subsections contain some more constructors for cat<span class="SimpleMath">\( <p><a id="X79944C7B87F767FD" name="X79944C7B87F767FD"></a></p> <h5>2.4-3 DiagonalCat1Group</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Di <p>This operation constructs examples of cat<span class="SimpleMath">\(^1\)</span>-groups of t <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">C4 := DiagonalCat1Group( [ (1,2,3), <span class="GAPprompt">gap></span> <span class="GAPinput">SetName( Source(C4), "a4a4" ); SetN <span class="GAPprompt">gap></span> <span class="GAPinput">Display( C4 );</span> Cat1-group [a4a4=>a4d] :- : Source group a4a4 has generators: [ (1,2,3), (2,3,4), (5,6,7), (6,7,8) ] : Range group a4d has generators: [ ( 9,10,11), (10,11,12) ] : tail homomorphism maps source generators to: [ ( 9,10,11), (10,11,12), (), () ] : head homomorphism maps source generators to: [ (), (), ( 9,10,11), (10,11,12) ] : range embedding maps range generators to: [ (1,2,3)(5,6,7), (2,3,4)(6,7,8) ] : kernel has generators: [ (5,6,7), (6,7,8) ] : boundary homomorphism maps generators of kernel to: [ ( 9,10,11), (10,11,12) ] : kernel embedding maps generators of kernel to: [ (5,6,7), (6,7,8) ] </pre></div> <p><a id="X79385660821E54A3" name="X79385660821E54A3"></a></p> <h5>2.4-4 TransposeCat1Group</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Tr <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Tr <p>The <em>transpose</em> of a cat<span class="SimpleMath">\(^1\)</span>-group <span class="Simple <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">R4 := TransposeCat1Group( C4 );</spa [a4a4=>a4d] <span class="GAPprompt">gap></span> <span class="GAPinput">Boundary( R4 );</span> [ (2,3,4), (1,2,3) ] -> [ (10,11,12), (9,10,11) ] <span class="GAPprompt">gap></span> <span class="GAPinput">TailMap( R4 ) = HeadMap( R4 ); </span> false <span class="GAPprompt">gap></span> <span class="GAPinput">TailMap( R4 ) = HeadMap( C4 ); </span> true <span class="GAPprompt">gap></span> <span class="GAPinput">MappingGeneratorsImages( Transpo [ [ [ (1,2,3), (2,3,4), (5,6,7), (6,7,8) ], [ (5,6,7), (6,7,8), (1,2,3), (2,3,4) ] ], [ [ (9,10,11), (10,11,12) ], [ (9,10,11), (10,11,12) ] ] ] </pre></div> <p><a id="X87544FAD873672E1" name="X87544FAD873672E1"></a></p> <h5>2.4-5 Cat1GroupByPeifferQuotient</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ca <p>If <span class="SimpleMath">\(C = (e;t,h : G \to R)\)</span> is a pre-cat<span class="SimpleMath <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">s4 := Group( (1,2,3), (3,4) );; SetNa <span class="GAPprompt">gap></span> <span class="GAPinput">h := GroupHomomorphismByImages( s4 <span class="GAPprompt">gap></span> <span class="GAPinput">c2 := Image( h );; SetName( c2, "c2" );< <span class="GAPprompt">gap></span> <span class="GAPinput">C := PreCat1Group( h, h );</span> [s4=>c2] <span class="GAPprompt">gap></span> <span class="GAPinput">P := PeifferSubgroupPreCat1Group( Group([ (1,3)(2,4), (1,2)(3,4) ]) <span class="GAPprompt">gap></span> <span class="GAPinput">C2 := Cat1GroupByPeifferQuotient( [Group( [ f1, f2 ] )=>c2] <span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription( C2 );</span> [ "S3", "C2" ] <span class="GAPprompt">gap></span> <span class="GAPinput">rec2 := PreXModRecordOfPreCat1Gro <span class="GAPprompt">gap></span> <span class="GAPinput">XC := rec2.prexmod;;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription( XC ); </span> [ "A4", "C2" ] <span class="GAPprompt">gap></span> <span class="GAPinput">XC2 := XModByPeifferQuotient( XC ); <span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription( XC2 );</span> [ "C3", "C2" ] <span class="GAPprompt">gap></span> <span class="GAPinput">CXC2 := Cat1GroupOfXMod( XC2 );;</sp <span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription( CXC2 );</spa [ "S3", "C2" ] <span class="GAPprompt">gap></span> <span class="GAPinput">IsomorphismCat1Groups( C2, CXC2 );< [[Group( [ f1, f2 ] ) => c2] => [(..|X..) => c2]] </pre></div> <p><a id="X85D9C5F881DBA9FC" name="X85D9C5F881DBA9FC"></a></p> <h5>2.4-6 SubCat1Group</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Su <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Su <p><span class="SimpleMath">\(S_1\)</span> is a sub-cat<span class="SimpleMath">\(^1\)</span>-g |
