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<p><a id="X8339DF98872D2E1C" name="X8339DF98872D2E1C"></a></p>
<div class="ChapSects"><a href="chap7.html#X8339DF98872D2E1C">7 <span class="Heading">Induced constructions</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X80D3C8F97A10D5E5">7.1 <span class="Heading">Coproducts of crossed modules</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X7C01F5D98046E44B">7.1-1 CoproductXMod</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X7966FF497C36C465">7.2 <span class="Heading">Induced crossed modules</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X874CB2A278AADE3A">7.2-1 InducedXMod</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X7B24D47F8078540F">7.2-2 AllInducedXMods</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X814A695779706E22">7.3 <span class="Heading">Induced cat<span class="SimpleMath">^1</span>-groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X7BCE57BE7F6E6B08">7.3-1 InducedCat1Group</a></span>
</div></div>
</div>
<h3>7 <span class="Heading">Induced constructions</span></h3>
<p>Before describing general functions for computing induced structures, we consider coproducts of crossed modules which provide induced crossed modules in certain cases.</p>
<p><a id="X80D3C8F97A10D5E5" name="X80D3C8F97A10D5E5"></a></p>
<h4>7.1 <span class="Heading">Coproducts of crossed modules</span></h4>
<p>Need to add here a reference (or two) for coproducts.</p>
<p><a id="X7C01F5D98046E44B" name="X7C01F5D98046E44B"></a></p>
<h5>7.1-1 CoproductXMod</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CoproductXMod</code>( <var class="Arg">X1</var>, <var class="Arg">X2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CoproductInfo</code>( <var class="Arg">X0</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>This function calculates the coproduct crossed module of two or more crossed modules which have a common range <span class="SimpleMath">R</span>. The standard method applies to <span class="SimpleMath">calX_1 = (∂_1 : S_1 -> R)</span> and <span class="SimpleMath">calX_2 = (∂_2 : S_2 -> R)</span>. See below for the case of three or more crossed modules.</p>
<p>The source <span class="SimpleMath">S_2</span> of <span class="SimpleMath">calX_2</span> acts on <span class="SimpleMath">S_1</span> via <span class="SimpleMath">∂_2</span> and the action of <span class="SimpleMath">calX_1</span>, so we can form a precrossed module <span class="SimpleMath">(∂' : S_1 ⋉ S_2 -> R)</span> where <span class="SimpleMath">∂'(s_1,s_2) = (∂_1 s_1)(∂_2 s_2)</span>. The action of this precrossed module is the diagonal action <span class="SimpleMath">(s_1,s_2)^r = (s_1^r,s_2^r)</span>. Factoring out by the Peiffer subgroup, we obtain the coproduct crossed module <span class="SimpleMath">calX_1 ∘ calX_2</span>.</p>
<p>In the example the structure descriptions of the precrossed module, the Peiffer subgroup, and the resulting coproduct are printed out when <code class="code">InfoLevel(InfoXMod)</code> is at least <span class="SimpleMath">1</span>. The coproduct comes supplied with attribute <code class="code">CoproductInfo</code>, which includes the embedding morphisms of the two factors.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">q8 := Group( (1,2,3,4)(5,8,7,6), (1,5,3,7)(2,6,4,8) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">XAq8 := XModByAutomorphismGroup( q8 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">s4b := Range( XAq8 );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetName( q8, "q8" ); SetName( s4b, "s4b" ); </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">a := q8.1;; b := q8.2;; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">alpha := GroupHomomorphismByImages( q8, q8, [a,b], [a^-1,b] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">beta := GroupHomomorphismByImages( q8, q8, [a,b], [a,b^-1] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">k4b := Subgroup( s4b, [ alpha, beta ] );; SetName( k4b, "k4b" );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Z8 := XModByNormalSubgroup( s4b, k4b );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetName( XAq8, "XAq8" ); SetName( Z8, "Z8" ); </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetInfoLevel( InfoXMod, 1 ); </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">XZ8 := CoproductXMod( XAq8, Z8 );</span>
#I prexmod is [ [ 32, 47 ], [ 24, 12 ] ]
#I peiffer subgroup is C2, [ 2, 1 ]
#I the coproduct is [ "C2 x C2 x C2 x C2", "S4" ], [ [ 16, 14 ], [ 24, 12 ] ]
[Group( [ f1, f2, f3, f4 ] )->s4b]
<span class="GAPprompt">gap></span> <span class="GAPinput">SetName( XZ8, "XZ8" ); </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">info := CoproductInfo( XZ8 );</span>
rec( embeddings := [ [XAq8 => XZ8], [Z8 => XZ8] ], xmods := [ XAq8, Z8 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">SetInfoLevel( InfoXMod, 0 ); </span>
</pre></div>
<p>Given a list of more than two crossed modules with a common range <span class="SimpleMath">R</span>, then an iterated coproduct is formed:</p>
<p class="pcenter">
\bigcirc~\left[ \calX_1,\calX_2,\ldots,\calX_n\right]
~=~ \calX_1 \circ (\calX_2 \circ ( \ldots
(\calX_{n-1} \circ \calX_n) \ldots ) ).
</p>
<p>The <code class="code">embeddings</code> field of the <code class="code">CoproductInfo</code> of the resulting crossed module <span class="SimpleMath">calY</span> contains the <span class="SimpleMath">n</span> morphisms <span class="SimpleMath">ϵ_i : calX_i -> calY (1 leqslant i leqslant n)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Y := CoproductXMod( [ XAq8, XAq8, Z8, Z8 ] );</span>
[Group( [ f1, f2, f3, f4, f5, f6, f7, f8 ] )->s4b]
<span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription( Y ); </span>
[ "C2 x C2 x C2 x C2 x C2 x C2 x C2 x C2", "S4" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">CoproductInfo( Y );</span>
rec(
embeddings :=
[ [XAq8 => [Group( [ f1, f2, f3, f4, f5, f6, f7, f8 ] ) -> s4b]],
[XAq8 => [Group( [ f1, f2, f3, f4, f5, f6, f7, f8 ] ) -> s4b]],
[Z8 => [Group( [ f1, f2, f3, f4, f5, f6, f7, f8 ] ) -> s4b]],
[Z8 => [Group( [ f1, f2, f3, f4, f5, f6, f7, f8 ] ) -> s4b]] ],
xmods := [ XAq8, XAq8, Z8, Z8 ] )
</pre></div>
<p><a id="X7966FF497C36C465" name="X7966FF497C36C465"></a></p>
<h4>7.2 <span class="Heading">Induced crossed modules</span></h4>
<p><a id="X874CB2A278AADE3A" name="X874CB2A278AADE3A"></a></p>
<h5>7.2-1 InducedXMod</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ InducedXMod</code>( <var class="Arg">args</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsInducedXMod</code>( <var class="Arg">xmod</var> )</td><td class="tdright">( property )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ InducedXModBySurjection</code>( <var class="Arg">xmod</var>, <var class="Arg">hom</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ InducedXModByCopower</code>( <var class="Arg">xmod</var>, <var class="Arg">hom</var>, <var class="Arg">list</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MorphismOfInducedXMod</code>( <var class="Arg">xmod</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>A morphism of crossed modules <span class="SimpleMath">(σ, ρ) : calX_1 -> calX_2</span> factors uniquely through an induced crossed module <span class="SimpleMath">ρ_∗ calX_1 = (δ : ρ_∗ S_1 -> R_2)</span>. Similarly, a morphism of cat<span class="SimpleMath">^1</span>-groups factors through an induced cat<span class="SimpleMath">^1</span>-group. Calculation of induced crossed modules of <span class="SimpleMath">calX</span> also provides an algebraic means of determining the homotopy <span class="SimpleMath">2</span>-type of homotopy pushouts of the classifying space of <span class="SimpleMath">calX</span>. For more background from algebraic topology see references in <a href="chapBib.html#biBBH1">[BH78]</a>, <a href="chapBib.html#biBBW1">[BW95]</a>, <a href="chapBib.html#biBBW2">[BW96]</a>. Induced crossed modules and induced cat<span class="SimpleMath">^1</span>-groups also provide the building blocks for constructing pushouts in the categories <em>XMod</em> and <em>Cat1</em>.</p>
<p>Data for the cases of algebraic interest is provided by a crossed module <span class="SimpleMath">calX = (∂ : S -> R)</span> and a homomorphism <span class="SimpleMath">ι : R -> Q</span>. The output from the calculation is a crossed module <span class="SimpleMath">ι_∗calX = (δ : ι_∗S -> Q)</span> together with a morphism of crossed modules <span class="SimpleMath">calX -> ι_∗calX</span>. When <span class="SimpleMath">ι</span> is a surjection with kernel <span class="SimpleMath">K</span> then <span class="SimpleMath">ι_∗S = S/[K,S]</span> where <span class="SimpleMath">[K,S]</span> is the subgroup of <span class="SimpleMath">S</span> generated by elements of the form <span class="SimpleMath">s^-1s^k, s ∈ S, k ∈ K</span> (see <a href="chapBib.html#biBBH1">[BH78]</a>, Prop.9). (For many years, up until June 2018, this manual has stated the result to be <span class="SimpleMath">[K,S]</span>, though the correct quotient had been calculated.) When <span class="SimpleMath">ι</span> is an inclusion the induced crossed module may be calculated using a copower construction <a href="chapBib.html#biBBW1">[BW95]</a> or, in the case when <span class="SimpleMath">R</span> is normal in <span class="SimpleMath">Q</span>, as a coproduct of crossed modules (<a href="chapBib.html#biBBW2">[BW96]</a>, but not yet implemented). When <span class="SimpleMath">ι</span> is neither a surjection nor an inclusion, <span class="SimpleMath">ι</span> is factored as the composite of the surjection onto the image and the inclusion of the image in <span class="SimpleMath">Q</span>, and then the composite induced crossed module is constructed. These constructions use Tietze transformation routines in the library file <code class="code">tietze.gi</code>.</p>
<p>As a first, surjective example, we take for <span class="SimpleMath">calX</span> a central extension crossed module of dihedral groups, <span class="SimpleMath">(d_24 -> d_12)</span>, and for <span class="SimpleMath">ι</span> a surjection <span class="SimpleMath">d_12 -> s_3</span> with kernel <span class="SimpleMath">c_2</span>. The induced crossed module is isomorphic to <span class="SimpleMath">(d_12 -> s_3)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">a := (6,7,8,9)(10,11,12);; b := (7,9)(11,12);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">d24 := Group( [ a, b ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetName( d24, "d24" );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">c := (1,2)(3,4,5);; d := (4,5);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">d12 := Group( [ c, d ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetName( d12, "d12" );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">bdy := GroupHomomorphismByImages( d24, d12, [a,b], [c,d] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">X24 := XModByCentralExtension( bdy );</span>
[d24->d12]
<span class="GAPprompt">gap></span> <span class="GAPinput">e := (13,14,15);; f := (14,15);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">s3 := Group( [ e, f ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetName( s3, "s3" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">epi := GroupHomomorphismByImages( d12, s3, [c,d], [e,f] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">iX24 := InducedXModBySurjection( X24, epi );</span>
[d24/ker->s3]
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( iX24 ); </span>
Crossed module [d24/ker->s3] :-
: Source group d24/ker has generators:
[ ( 1,11, 5, 4,10, 8)( 2,12, 6, 3, 9, 7),
( 1, 2)( 3, 4)( 5, 9)( 6,10)( 7,11)( 8,12) ]
: Range group s3 has generators:
[ (13,14,15), (14,15) ]
: Boundary homomorphism maps source generators to:
[ (13,14,15), (14,15) ]
: Action homomorphism maps range generators to automorphisms:
(13,14,15) --> { source gens --> [ ( 1,11, 5, 4,10, 8)( 2,12, 6, 3, 9, 7),
( 1, 6)( 2, 5)( 3, 8)( 4, 7)( 9,10)(11,12) ] }
(14,15) --> { source gens --> [ ( 1, 8,10, 4, 5,11)( 2, 7, 9, 3, 6,12),
( 1, 2)( 3, 4)( 5, 9)( 6,10)( 7,11)( 8,12) ] }
These 2 automorphisms generate the group of automorphisms.
<span class="GAPprompt">gap></span> <span class="GAPinput">morX24 := MorphismOfInducedXMod( iX24 );</span>
[[d24->d12] => [d24/ker->s3]]
</pre></div>
<p>For a second, injective example we take for <span class="SimpleMath">calX</span> the result <code class="code">iX24</code> of the previous example and for <span class="SimpleMath">ι</span> an inclusion of <span class="SimpleMath">s_3</span> in <span class="SimpleMath">s_4</span>. The resulting source group has size <span class="SimpleMath">96</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g := (16,17,18);; h := (16,17,18,19);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">s4 := Group( [ g, h ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetName( s4, "s4" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">iota := GroupHomomorphismByImages( s3, s4, [e,f], [g^2*h^2,g*h^-1] );</span>
[ (13,14,15), (14,15) ] -> [ (17,18,19), (18,19) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">iiX24 := InducedXModByCopower( iX24, iota, [ ] );</span>
i*([d24/ker->s3])
<span class="GAPprompt">gap></span> <span class="GAPinput">Size2d( iiX24 ); </span>
[ 96, 24 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription( iiX24 );</span>
[ "C2 x GL(2,3)", "S4" ]
</pre></div>
<p>For a third example we combine the previous two examples by taking for <span class="SimpleMath">ι</span> the more general case <code class="code">alpha = theta*iota</code>. The resulting <code class="code">jX24</code> is isomorphic to, but not identical to, <code class="code">iiX24</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">alpha := CompositionMapping( iota, epi );</span>
[ (1,2)(3,4,5), (4,5) ] -> [ (17,18,19), (18,19) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">jX24 := InducedXMod( X24, alpha );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription( jX24 );</span>
[ "C2 x GL(2,3)", "S4" ]
</pre></div>
<p>For a fourth example we use the version <code class="code">InducedXMod(Q,R,S)</code> of this global function, with a normal inclusion crossed module <span class="SimpleMath">(S -> R)</span> and an inclusion mapping <span class="SimpleMath">R -> Q</span>. We take <span class="SimpleMath">(c_6 -> d_12)</span> as <span class="SimpleMath">calX</span> and the inclusion of <span class="SimpleMath">d_12</span> in <span class="SimpleMath">d_24</span> as <span class="SimpleMath">ι</span>.</p>
<div class="example"><pre>
## Section 7.2.1 : Example 4
<span class="GAPprompt">gap></span> <span class="GAPinput">d12b := Subgroup( d24, [ a^2, b ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetName( d12b, "d12b" ); </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">c6b := Subgroup( d12b, [ a^2 ] );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetName( c6b, "c6b" ); </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">X12 := InducedXMod( d24, d12b, c6b );</span>
i*([c6b->d12b])
<span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription( X12 );</span>
[ "C6 x C6", "D24" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( MorphismOfInducedXMod( X12 ) );</span>
Morphism of crossed modules :-
: Source = [c6b->d12b] with generating sets:
[ ( 6, 8)( 7, 9)(10,12,11) ]
[ ( 6, 8)( 7, 9)(10,12,11), ( 7, 9)(11,12) ]
: Range = i*([c6b->d12b]) with generating sets:
[ ( 4, 5)( 6, 7)( 8, 9)(10,11)(12,13)(14,15),
( 4, 6, 8)( 5, 7, 9)(10,12,14)(11,13,15),
( 4,10)( 5,11)( 6,12)( 7,13)( 8,14)( 9,15), (1,2,3) ]
[ ( 6, 7, 8, 9)(10,11,12), ( 7, 9)(11,12) ]
: Source Homomorphism maps source generators to:
[ ( 4, 9, 6, 5, 8, 7)(10,15,12,11,14,13) ]
: Range Homomorphism maps range generators to:
[ ( 6, 8)( 7, 9)(10,12,11), ( 7, 9)(11,12) ]
#
</pre></div>
<p><a id="X7B24D47F8078540F" name="X7B24D47F8078540F"></a></p>
<h5>7.2-2 AllInducedXMods</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AllInducedXMods</code>( <var class="Arg">Q</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>This function calculates all the induced crossed modules <code class="code">InducedXMod(Q,R,S)</code>, where <code class="code">R</code> runs over all conjugacy classes of subgroups of <code class="code">Q</code> and <code class="code">S</code> runs over all non-trivial normal subgroups of <code class="code">R</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">all := AllInducedXMods( q8 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">L := List( all, x -> Source( x ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Sort( L, function(g,h) return Size(g) < Size(h); end );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">List( L, x -> StructureDescription( x ) );</span>
[ "1", "1", "1", "1", "C2 x C2", "C2 x C2", "C2 x C2", "C4 x C4", "C4 x C4",
"C4 x C4", "C2 x C2 x C2 x C2" ]
</pre></div>
<p><a id="X814A695779706E22" name="X814A695779706E22"></a></p>
<h4>7.3 <span class="Heading">Induced cat<span class="SimpleMath">^1</span>-groups</span></h4>
<p><a id="X7BCE57BE7F6E6B08" name="X7BCE57BE7F6E6B08"></a></p>
<h5>7.3-1 InducedCat1Group</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ InducedCat1Group</code>( <var class="Arg">args</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ InducedCat1GroupByFreeProduct</code>( <var class="Arg">grp</var>, <var class="Arg">hom</var> )</td><td class="tdright">( property )</td></tr></table></div>
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