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</span><a href="chap8_mj.html </span> <div class="ContSSBlock"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X7D421DE57B44F37 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X783F7FEB7B79B06 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X84636E047CDF2DF <span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X7EFCF9697E845B2 </div></div> </div> <h3>8 <span class="Heading">Crossed squares and Cat<span class="SimpleMath">\(^2\)</span>-grou <p>The term <em>3d-group</em> refers to a set of equivalent categories of which the most common are th <p>The material in this chapter should be considered experimental. A major overhaul took place i <p><a id="X7C4AFE8D85848C8F" name="X7C4AFE8D85848C8F"></a></p> <h4>8.1 <span class="Heading">Definition of a crossed square and a crossed <span class="SimpleMath">\(n\)</span>-cube of groups</span></h4> <p>Crossed squares were introduced by Guin-Waléry and Loday (see, for example, <a href="chapBib_ <p>A <em>crossed square</em> <span class="SimpleMath">\(\calS\)</span> consists of the following:</ <ul> <li><p>groups <span class="SimpleMath">\(S_J\)</span> for each of the four subsets <span class="Sim </li> <li><p>a commutative diagram of group homomorphisms:</p> <p class="center">\[ \ddot{\partial}_1 : S_{[2]} \to S_{\{2\}}, \quad \ddot{\partial}_2 : S_{[2]} \to S_{\{1\}}, \quad \dot{\partial}_2 : S_{\{2\}} \to S_{\emptyset}, \quad \dot{\partial}_1 : S_{\{1\}} \to S_{\emptyset} \]</p> <p>(again we often write <span class="SimpleMath">\(\kappa = \ddot{\partial}_1,~ \lambda = \dd </li> <li><p>actions of <span class="SimpleMath">\(S_{\emptyset}\)</span> on <span class="SimpleMath"> </li> <li><p>a function <span class="SimpleMath">\(\boxtimes : S_{\{1\}} \times S_{\{2\}} \to S_{[2]} </li> </ul> <p>Here is a picture of the situation:</p> <p class="center">\[ \vcenter{\xymatrix{ & & S_{[2]} \ar[rr]^{\ddot{\partial}_1} \ar[dd]_{\ddot{\partial}_2} && S_{\{2\}} \ar[dd]^{\dot{\partial}_2} && L \ar[rr]^{\kappa} \ar[dd]_{\lambda} && M \ar[dd]^{\mu} & \\ \mathcal{S} & = & && & = && \\ & & S_{\{1\}} \ar[rr]_{\dot{\partial}_1} && S_{\emptyset} && N \ar[rr]_{\nu} && P }} \]</p> <p>The following axioms must be satisfied for all <span class="SimpleMath">\(l \in L,\; m,m_1,m_ <ul> <li><p>The homomorphisms <span class="SimpleMath">\(\kappa, \lambda\)</span> preserve the actio </li> <li><p>Each of the upper, left-hand, right-hand and lower sides of the square,</p> <p class="center">\[ \ddot{\calS}_1 = (\kappa : L \to M), \quad \ddot{\calS}_2 = (\lambda : L \to N), \quad \dot{\calS}_2 = (\mu : M \to P), \quad \dot{\calS}_1 = (\nu : N \to P), \]</p> <p>and the diagonal</p> <p class="center">\[ \calS_{12} = (\partial_{12} := \mu \circ \kappa = \nu \circ \lambda : L \to P) \]</p> <p>are crossed modules (with actions via <span class="SimpleMath">\(P\)</span>).</p> <p>These will be called the <em>up, left, right, down</em> and <em>diagonal</em> crossed modules of <span </li> <li><p><span class="SimpleMath">\(\boxtimes\)</span> is a <em>crossed pairing</em>:</p> <ul> <li><p><span class="SimpleMath">\((n_1n_2 \boxtimes m)\;=\; {(n_1 \boxtimes m)}^{n_2}\;(n_2 \ </li> <li><p><span class="SimpleMath">\((n \boxtimes m_1m_2) \;=\; (n \boxtimes m_2)\;{(n \boxtimes m </li> <li><p><span class="SimpleMath">\((n \boxtimes m)^{p} \;=\; (n^p \boxtimes m^p)\)</span>.</p> </li> </ul> </li> <li><p><span class="SimpleMath">\(\kappa(n \boxtimes m) \;=\; (m^{-1})^{n}\;m \quad \mbox{and </li> <li><p><span class="SimpleMath">\((n \boxtimes \kappa l) \;=\; (l^{-1})^{n}\;l \quad \mbox{and </li> </ul> <p>Note that the actions of <span class="SimpleMath">\(M\)</span> on <span class="SimpleMath">\(N <p class="center">\[ {n_1}^{(m^n)} \;=\; {n_1}^{\mu(m^n)} \;=\; {n_1}^{n^{-1}(\mu m)n} \;=\; (({n_1}^{n^{-1}})^m)^n. \]</p> <p>A <em>precrossed square</em> is a similar structure which satisfies some subset of these axioms. <p>Crossed squares are the <span class="SimpleMath">\(n=2\)</span> case of a crossed <span class=" <p>A <em>crossed</em> <span class="SimpleMath">\(n\)</span><em>-cube of groups</em> consists of the fol <ul> <li><p>groups <span class="SimpleMath">\(S_A\)</span> for every subset <span class="SimpleMath">\ </li> <li><p>a commutative diagram of group homomorphisms <span class="SimpleMath">\(\partial_i : S_A </li> <li><p>actions of <span class="SimpleMath">\(S_{\emptyset}\)</span> on each <span class="SimpleM </li> <li><p>functions <span class="SimpleMath">\(\boxtimes_{A,B} : S_A \times S_B \to S_{A \cup B}, (A </li> </ul> <p>There is then a long list of axioms which must be satisfied.</p> <p><a id="X820A7D30847BC828" name="X820A7D30847BC828"></a></p> <h4>8.2 <span class="Heading">Constructions for crossed squares</span></h4> <p>Analogously to the data structure used for crossed modules, crossed squares are implemented <p><a id="X866A7FAC7FCB62C2" name="X866A7FAC7FCB62C2"></a></p> <h5>8.2-1 CrossedSquareByXMods</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Cr <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Pr <p>If <em>up,left,right,down,diag</em> are five (pre-)crossed modules whose sources and ranges a <p>The <code class="code">Display</code> function is used to print details of 3d-groups.</p> <p>We take as our example a simple, but significant case. We start with five crossed modules forme <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">b := (2,4);; c := (1,2)(3,4);; p := (1, <span class="GAPprompt">gap></span> <span class="GAPinput">d8 := Group( b, c );; </span> <span class="GAPprompt">gap></span> <span class="GAPinput">SetName( d8, "d8" );; </span> <span class="GAPprompt">gap></span> <span class="GAPinput">L := Subgroup( d8, [p^2] );; </span> <span class="GAPprompt">gap></span> <span class="GAPinput">M := Subgroup( d8, [b] );; </span> <span class="GAPprompt">gap></span> <span class="GAPinput">N := Subgroup( d8, [c] );; </span> <span class="GAPprompt">gap></span> <span class="GAPinput">P := TrivialSubgroup( d8 );; </span> <span class="GAPprompt">gap></span> <span class="GAPinput">kappa := GroupHomomorphismByImage <span class="GAPprompt">gap></span> <span class="GAPinput">lambda := GroupHomomorphismByImag <span class="GAPprompt">gap></span> <span class="GAPinput">delta := GroupHomomorphismByImage <span class="GAPprompt">gap></span> <span class="GAPinput">mu := GroupHomomorphismByImages( M <span class="GAPprompt">gap></span> <span class="GAPinput">nu := GroupHomomorphismByImages( N <span class="GAPprompt">gap></span> <span class="GAPinput">up := XModByTrivialAction( kappa ); <span class="GAPprompt">gap></span> <span class="GAPinput">left := XModByTrivialAction( lambd <span class="GAPprompt">gap></span> <span class="GAPinput">diag := XModByTrivialAction( delta <span class="GAPprompt">gap></span> <span class="GAPinput">right := XModByTrivialAction( mu ); <span class="GAPprompt">gap></span> <span class="GAPinput">down := XModByTrivialAction( nu );; < <span class="GAPprompt">gap></span> <span class="GAPinput">xp := CrossedPairingByCommutators <span class="GAPprompt">gap></span> <span class="GAPinput">Print( "xp([c,b]) = ", xp( c, b ), "\n" xp([c,b]) = (1,3)(2,4) <span class="GAPprompt">gap></span> <span class="GAPinput">PXS := PreCrossedSquareByPreXMods <span class="GAPprompt">gap></span> <span class="GAPinput">Display( PXS ); </span> (pre-)crossed square with (pre-)crossed modules: up = [Group( [ (1,3)(2,4) ] ) -> Group( [ (2,4) ] )] left = [Group( [ (1,3)(2,4) ] ) -> Group( [ (1,2)(3,4) ] )] right = [Group( [ (2,4) ] ) -> Group( () )] down = [Group( [ (1,2)(3,4) ] ) -> Group( () )] <span class="GAPprompt">gap></span> <span class="GAPinput">IsCrossedSquare( PXS ); </span> false </pre></div> <p><a id="X7FA98BD47FF9B044" name="X7FA98BD47FF9B044"></a></p> <h5>8.2-2 Size3d</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Si <p>Just as <code class="code">Size2d</code> was used in place of <code class="code">Size</code> for cr <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">Size3d( PXS ); </span> [ 2, 2, 2, 1 ] </pre></div> <p><a id="X7896DAF786F46234" name="X7896DAF786F46234"></a></p> <h5>8.2-3 CrossedSquareByNormalSubgroups</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Cr <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Cr <p>If <span class="SimpleMath">\(L, M, N\)</span> are normal subgroups of a group <span class="Simp <p class="center">\[ \boxtimes \;:\; N \times M \to L, \quad (n,m) \mapsto [n,m] \,=\, n^{-1}m^{-1}nm \,=\,(m^{-1})^nm \,=\, n^{-1}n^m\,. \]</p> <p>This construction is implemented as <code class="code">CrossedSquareByNormalSubgroups(L <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">d20 := DihedralGroup( IsPermGroup, <span class="GAPprompt">gap></span> <span class="GAPinput">gend20 := GeneratorsOfGroup( d20 ); < [ (1,2,3,4,5,6,7,8,9,10), (2,10)(3,9)(4,8)(5,7) ] <span class="GAPprompt">gap></span> <span class="GAPinput">p1 := gend20[1];; p2 := gend20[2];; p (1,10)(2,9)(3,8)(4,7)(5,6) <span class="GAPprompt">gap></span> <span class="GAPinput">d10a := Subgroup( d20, [ p1^2, p2 ] );;< <span class="GAPprompt">gap></span> <span class="GAPinput">d10b := Subgroup( d20, [ p1^2, p12 ] ); <span class="GAPprompt">gap></span> <span class="GAPinput">c5d := Subgroup( d20, [ p1^2 ] );;</spa <span class="GAPprompt">gap></span> <span class="GAPinput">SetName( d20, "d20" ); SetName( d10a <span class="GAPprompt">gap></span> <span class="GAPinput">SetName( d10b, "d10b" ); SetName( c5 <span class="GAPprompt">gap></span> <span class="GAPinput">XSconj := CrossedSquareByNormalSu [ c5d -> d10a ] [ | | ] [ d10b -> d20 ] <span class="GAPprompt">gap></span> <span class="GAPinput">xpc := CrossedPairing( XSconj );;</s <span class="GAPprompt">gap></span> <span class="GAPinput">xpc( p12, p2 );</span> (1,3,5,7,9)(2,4,6,8,10) </pre></div> <p><a id="X7FA367977B1895A7" name="X7FA367977B1895A7"></a></p> <h5>8.2-4 CrossedSquareByNormalSubXMod</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Cr <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Cr <p>If <span class="SimpleMath">\(\calX_1 = (\partial_1 : S_1 \to R_1)\)</span> is a normal sub-cro <p class="center">\[ \boxtimes \;:\; R_1 \times S_0 \to S_1, \quad (r_1,s_0) \mapsto (s_0^{-1})^{r_1} s_0. \]</p> <p>The example constructs the same crossed square as in the previous subsection.</p> <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">X20 := XModByNormalSubgroup( d20, d <span class="GAPprompt">gap></span> <span class="GAPinput">X10 := XModByNormalSubgroup( d10b, <span class="GAPprompt">gap></span> <span class="GAPinput">ok := IsNormalSub2DimensionalDoma true <span class="GAPprompt">gap></span> <span class="GAPinput">XS20 := CrossedSquareByNormalSubX [ c5d -> d10a ] [ | | ] [ d10b -> d20 ] <span class="GAPprompt">gap></span> <span class="GAPinput">xp20 := CrossedPairing( XS20 );; </sp <span class="GAPprompt">gap></span> <span class="GAPinput">xp20( p1^2, p2 );</span> (1,7,3,9,5)(2,8,4,10,6) </pre></div> <p><a id="X833362FE87ED3C48" name="X833362FE87ED3C48"></a></p> <h5>8.2-5 ActorCrossedSquare</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ac <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Cr <p>The actor <span class="SimpleMath">\(\calA(\calX_0)\)</span> of a crossed module <span class= <p class="center">\[ \boxtimes \;:\; R \times W \,\to\, S, \quad (r,\chi) \,\mapsto\, \chi r~. \]</p> <p>This is implemented as <code class="code">ActorCrossedSquare(X0)</code>.</p> <p>The example constructs <code class="code">XSact</code>, the actor crossed square of the crosse <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">XSact := ActorCrossedSquare( X20 );< crossed square with crossed modules: up = Whitehead[d10a->d20] left = [d10a->d20] right = Actor[d10a->d20] down = Norrie[d10a->d20] <span class="GAPprompt">gap></span> <span class="GAPinput">W := Range( Up2DimensionalGroup( XS Group([ (2,5)(3,4), (2,3,5,4), (1,4,2,5,3) ]) <span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription( W );</span> "C5 : C4" <span class="GAPprompt">gap></span> <span class="GAPinput">xpa := CrossedPairing( XSact );;</sp <span class="GAPprompt">gap></span> <span class="GAPinput">xpa( p1, (2,3,5,4) );</span> (1,7,3,9,5)(2,8,4,10,6) </pre></div> <p><a id="X82E4EA93824F7B26" name="X82E4EA93824F7B26"></a></p> <h5>8.2-6 CrossedSquareByAutomorphismGroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Cr <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Cr <p>For <span class="SimpleMath">\(G\)</span> a group let <span class="SimpleMath">\(\Inn(G)\)</s <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">AXS20 := CrossedSquareByAutomorph [ d20 -> Inn(d20) ] [ | | ] [ Inn(d20) -> Aut(d20) ] <span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription( AXS20 );</sp [ "D20", "D10", "D10", "C2 x (C5 : C4)" ] <span class="GAPprompt">gap></span> <span class="GAPinput">I20 := Range( Up2DimensionalGroup( <span class="GAPprompt">gap></span> <span class="GAPinput">genI20 := GeneratorsOfGroup( I20 ); < [ ^(1,2,3,4,5,6,7,8,9,10), ^(2,10)(3,9)(4,8)(5,7) ] <span class="GAPprompt">gap></span> <span class="GAPinput">xpi := CrossedPairing( AXS20 );;</sp <span class="GAPprompt">gap></span> <span class="GAPinput">xpi( genI20[1], genI20[2] );</span> (1,9,7,5,3)(2,10,8,6,4) </pre></div> <p><a id="X839E065783795CB8" name="X839E065783795CB8"></a></p> <h5>8.2-7 CrossedSquareByPullback</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Cr <p>If crossed modules <span class="SimpleMath">\(\calX_1 = (\nu : N \to P)\)</span> and <span class= <p class="center">\[ \boxtimes \;:\; N \times M \to L, \quad (n,m) \mapsto (n^{-1}n^{\mu m}, (m^{-1})^{\nu n}m) . \]</p> <p>The second example below uses the central extension crossed module <code class="code">X12=(D <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">dn := Down2DimensionalGroup( XScon <span class="GAPprompt">gap></span> <span class="GAPinput">rt := Right2DimensionalGroup( XSco <span class="GAPprompt">gap></span> <span class="GAPinput">XSP := CrossedSquareByPullback( dn [ (d10b x_d20 d10a) -> d10a ] [ | | ] [ d10b -> d20 ] <span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription( XSP ); </span> [ "C5", "D10", "D10", "D20" ] <span class="GAPprompt">gap></span> <span class="GAPinput">XS12 := CrossedSquareByPullback( X <span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription( XS12 ); </spa [ "C2 x C2 x S3", "D12", "D12", "S3" ] <span class="GAPprompt">gap></span> <span class="GAPinput">xp12 := CrossedPairing( XS12 );; </sp <span class="GAPprompt">gap></span> <span class="GAPinput">xp12( (1,2,3,4,5,6), (2,6)(3,5) );< (1,5,3)(2,6,4)(7,11,9)(8,12,10) </pre></div> <p><a id="X7F5554907AF73190" name="X7F5554907AF73190"></a></p> <h5>8.2-8 CrossedSquareByXModSplitting</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Cr <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Cr <p>For <span class="SimpleMath">\(\calX = (\partial : S \to R)\)</span> let <span class="SimpleMat <p>A particular case is when <span class="SimpleMath">\(S\)</span> is an <span class="SimpleMath"> <p class="center">\[ \vcenter{\xymatrix{ & & S \ar[rr]^{\partial'} \ar[dd]_{\partial'} \ar[rrdd]^{\partial} && Q \ar[dd]^{\iota} && A \ar[rr]^0 \ar[dd]_0 \ar[rrdd]^0 && 1 \ar[dd]^{\iota} & \\ & & && & && \\ & & Q \ar[rr]_{\iota} && R && 1 \ar[rr]_{\iota} && R }} \]</p> <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">k4 := Group( (1,2), (3,4) );;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">AX4 := XModByAutomorphismGroup( k4 <span class="GAPprompt">gap></span> <span class="GAPinput">X4 := Image( IsomorphismPermObject <span class="GAPprompt">gap></span> <span class="GAPinput">XSS4 := CrossedSquareByXModSplitt <span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription( XSS4 );</spa [ "C2 x C2", "1", "1", "S3" ] <span class="GAPprompt">gap></span> <span class="GAPinput">XSS20 := CrossedSquareByXModSplit <span class="GAPprompt">gap></span> <span class="GAPinput">up20 := Up2DimensionalGroup( XSS20 <span class="GAPprompt">gap></span> <span class="GAPinput">Range( up20 ) = d10a; </span> true <span class="GAPprompt">gap></span> <span class="GAPinput">SetName( Range( up20 ), "d10a" ); </sp <span class="GAPprompt">gap></span> <span class="GAPinput">Name( XSS20 );;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">XSS20;</span> [d10a->d10a,d10a->d20] <span class="GAPprompt">gap></span> <span class="GAPinput">xps := CrossedPairing( XSS20 );;</sp <span class="GAPprompt">gap></span> <span class="GAPinput">xps( p1^2, p2 );</span> (1,7,3,9,5)(2,8,4,10,6) </pre></div> <p><a id="X87FBE3CE87DC8CD5" name="X87FBE3CE87DC8CD5"></a></p> <h5>8.2-9 CrossedSquare</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Cr <p>The function <code class="code">CrossedSquare</code> may be used to call some of the constructi <ul> <li><p><code class="code">CrossedSquare(X0)</code> calls <code class="code">CrossedSquareByXMo </li> <li><p><code class="code">CrossedSquare(C0)</code> calls <code class="code">CrossedSquareOfCat </li> <li><p><code class="code">CrossedSquare(X0,X1)</code> calls <code class="code">CrossedSquareBy </li> <li><p><code class="code">CrossedSquare(X0,X1)</code> calls <code class="code">CrossedSquareBy </li> <li><p><code class="code">CrossedSquare(L,M,N,P)</code> calls <code class="code">CrossedSquare </li> </ul> <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">diag := Diagonal2DimensionalGroup [d20->Aut(d20)] <span class="GAPprompt">gap></span> <span class="GAPinput">XSdiag := CrossedSquare( diag );; </s <span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription( XSdiag ); </s [ "D20", "D10", "D10", "C2 x (C5 : C4)" ] </pre></div> <p><a id="X7F94830681EA19BE" name="X7F94830681EA19BE"></a></p> <h5>8.2-10 Transpose3DimensionalGroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Tr <p>The <em>transpose</em> of a crossed square <span class="SimpleMath">\(\calS\)</span> is the cross <p class="center">\[ \tilde{\boxtimes} \;:\; M \times N \to L, \quad (m,n) \;\mapsto\; m\,\tilde{\boxtimes}\,n := (n \boxtimes m)^{-1}~. \]</p> <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">XStrans := Transpose3DimensionalG [ c5d -> d10b ] [ | | ] [ d10a -> d20 ] </pre></div> <p><a id="X7C2647CB82DDD065" name="X7C2647CB82DDD065"></a></p> <h5>8.2-11 CentralQuotient</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ce <p>The central quotient of a crossed module <span class="SimpleMath">\(\calX = (\partial : S \to R <ul> <li><p>the left crossed module is <span class="SimpleMath">\(\calX\)</span>;</p> </li> <li><p>the right crossed module is the quotient <span class="SimpleMath">\(\calX/Z(\calX)\)</sp </li> <li><p>the up and down homomorphisms are the natural homomorphisms onto the quotient groups;</p> </li> <li><p>the crossed pairing <span class="SimpleMath">\(\boxtimes : (R \times F) \to S\)</span>, wher </li> </ul> <p>This is the special case of an intended function <code class="code">CrossedSquareByCentralE <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">pos7 := Position( ids, [ [12,2], [24, <span class="GAPprompt">gap></span> <span class="GAPinput">Xn7 := nsx[pos7];; </span> <span class="GAPprompt">gap></span> <span class="GAPinput">IdGroup( Xn7 );</span> [ [ 12, 2 ], [ 24, 5 ] ] <span class="GAPprompt">gap></span> <span class="GAPinput">IdGroup( CentreXMod( Xn7 ) ); </span> [ [ 4, 1 ], [ 4, 1 ] ] <span class="GAPprompt">gap></span> <span class="GAPinput">CQXn7 := CentralQuotient( Xn7 );;</s <span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription( CQXn7 );</sp [ "C12", "C3", "C4 x S3", "S3" ] </pre></div> <p><a id="X8645AA3686F126D5" name="X8645AA3686F126D5"></a></p> <h5>8.2-12 IsCrossedSquare</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>These are the basic properties for 3d-groups, and crossed squares in particular.</p> <p><a id="X828CFC5A83097189" name="X828CFC5A83097189"></a></p> <h5>8.2-13 Up2DimensionalGroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Up <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Le <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Do <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ri <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Cr <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Di <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Na <p>These are the basic attributes of a crossed square <span class="SimpleMath">\(\calS\)</span>. <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">Up2DimensionalGroup( XSconj );</sp [c5d->d10a] <span class="GAPprompt">gap></span> <span class="GAPinput">Right2DimensionalGroup( XSact );</ Actor[d10a->d20] <span class="GAPprompt">gap></span> <span class="GAPinput">Name( XSconj ); </span> "[c5d->d10a,d10b->d20]" <span class="GAPprompt">gap></span> <span class="GAPinput">cross1 := CrossDiagonalActions( XS <span class="GAPprompt">gap></span> <span class="GAPinput">gensa := GeneratorsOfGroup( d10a ); <span class="GAPprompt">gap></span> <span class="GAPinput">gensb := GeneratorsOfGroup( d10a ); <span class="GAPprompt">gap></span> <span class="GAPinput">act1 := ImageElm( cross1, gensb[1] ) <span class="GAPprompt">gap></span> <span class="GAPinput">gensa[2]; ImageElm( act1, gensa[2] (2,10)(3,9)(4,8)(5,7) (1,5)(2,4)(6,10)(7,9) </pre></div> <p><a id="X7F81DA90820F4405" name="X7F81DA90820F4405"></a></p> <h5>8.2-14 IsSymmetric3DimensionalGroup</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>These are further properties for 3d-groups, and crossed squares in particular. A 3d-group is < <p><a id="X7AE671C7798F99FD" name="X7AE671C7798F99FD"></a></p> <h5>8.2-15 Crossed Pairing</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Cr <p>Crossed pairings have been implemented as functions of two variables, <span class="SimpleMa <p>The first example shows the crossed pairing in the crossed square <code class="code">XSconj</c <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">xp := CrossedPairing( XSconj );;</sp <span class="GAPprompt">gap></span> <span class="GAPinput">xp( (1,6)(2,5)(3,4)(7,10)(8,9), ( (1,7,8,5,3)(2,9,10,6,4) </pre></div> <p>The second example shows how to construct a crossed pairing.</p> <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">F := FreeGroup(1);;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">x := GeneratorsOfGroup(F)[1];;</sp <span class="GAPprompt">gap></span> <span class="GAPinput">z := GroupHomomorphismByImages( F, <span class="GAPprompt">gap></span> <span class="GAPinput">id := GroupHomomorphismByImages( F <span class="GAPprompt">gap></span> <span class="GAPinput">h := function(n,m)</span> <span class="GAPprompt">></span> <span class="GAPinput"> return x^(ExponentSumWord(n,x)*Expo <span class="GAPprompt">></span> <span class="GAPinput"> end;;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">h( x^3, x^4 );</span> f1^12 <span class="GAPprompt">gap></span> <span class="GAPinput">A := AutomorphismGroup( F );;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">a := GeneratorsOfGroup(A)[1];;</sp <span class="GAPprompt">gap></span> <span class="GAPinput">act := GroupHomomorphismByImages( <span class="GAPprompt">gap></span> <span class="GAPinput">X0 := XModByBoundaryAndAction( z, a <span class="GAPprompt">gap></span> <span class="GAPinput">X1 := XModByBoundaryAndAction( id, <span class="GAPprompt">gap></span> <span class="GAPinput">XSF := PreCrossedSquareByPreXMods <span class="GAPprompt">gap></span> <span class="GAPinput">IsCrossedSquare( XSF ); </span> true </pre></div> <p><a id="X79A59CED7C69BF18" name="X79A59CED7C69BF18"></a></p> <h4>8.3 <span class="Heading">Substructures of Crossed Squares</span></h4> <p>To specify a sub-crossed square of <span class="SimpleMath">\(\calS\)</span> we may specify fo <p><a id="X83F16E94857407F3" name="X83F16E94857407F3"></a></p> <h5>8.3-1 SubCrossedSquare</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">‣ Is <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">‣ Is <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">Display( XS20 ); </span> [ c5d -> d10a ] [ | | ] [ d10b -> d20 ] <span class="GAPprompt">gap></span> <span class="GAPinput">p5 := p1^5*p2;; [p2,p12,p5];</span> [ (2,10)(3,9)(4,8)(5,7), (1,10)(2,9)(3,8)(4,7)(5,6), (1,6)(2,5)(3,4)(7,10)(8,9) ] <span class="GAPprompt">gap></span> <span class="GAPinput">L1 := TrivialSubgroup( c5d );;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">M1 := Subgroup( d10a, [ p2 ] );;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">N1 := Subgroup( d10b, [ p5 ] );;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">P1 := Subgroup( d20, [ p2, p5 ] );;</spa <span class="GAPprompt">gap></span> <span class="GAPinput">sub20 := SubCrossedSquare( XS20, L1 crossed square with crossed modules: up = [Group( () ) -> Group( [ ( 2,10)( 3, 9)( 4, 8)( 5, 7) ] )] left = [Group( () ) -> Group( [ ( 1, 6)( 2, 5)( 3, 4)( 7,10)( 8, 9) ] )] right = [Group( [ ( 2,10)( 3, 9)( 4, 8)( 5, 7) ] ) -> Group( [ ( 2,10)( 3, 9)( 4, 8)( 5, 7), ( 1, 6)( 2, 5)( 3, 4)( 7,10)( 8, 9) ] )] down = [Group( [ ( 1, 6)( 2, 5)( 3, 4)( 7,10)( 8, 9) ] ) -> Group( [ ( 2,10)( 3, 9)( 4, 8)( 5, 7), ( 1, 6)( 2, 5)( 3, 4)( 7,10)( 8, 9) ] )] <span class="GAPprompt">gap></span> <span class="GAPinput">sxp := CrossedPairing( sub20 );;</sp <span class="GAPprompt">gap></span> <span class="GAPinput">sxp( p5, p2 );</span> () </pre></div> <p><a id="X7D0D7F787D438D9A" name="X7D0D7F787D438D9A"></a></p> <h5>8.3-2 TrivialSub3DimensionalGroup</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 <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Tr <p>A special case of the previous operation is when all the subgroups are trivial.</p> <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">TC2ab := TrivialSubCat2Group( C2ab (pre-)cat2-group with generating (pre-)cat1-groups: 1 : [Group( () ) => Group( () )] 2 : [Group( () ) => Group( () )] </pre></div> <p><a id="X78A79A7E85128C7B" name="X78A79A7E85128C7B"></a></p> <h4>8.4 <span class="Heading">Morphisms of crossed squares</span></h4> <p>This section describes an initial implementation of morphisms of (pre-)crossed squares.</p> <p><a id="X83E733547A14FD61" name="X83E733547A14FD61"></a></p> <h5>8.4-1 CrossedSquareMorphism</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Cr <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Cr <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Cr <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><a id="X7DE8173F80E07AB1" name="X7DE8173F80E07AB1"></a></p> <h5>8.4-2 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">‣ Up <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Le <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Do <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ri |
