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<div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0_mj.html">Top</a>  <a href="chap1_mj.html">1</a>  <a href="chap2_mj.html">2</a>  <a href="chap3_mj.html">3</a>  <a href="chap4_mj.html">4</a>  <a href="chap5_mj.html">5</a>  <a href="chap6_mj.html">6</a>  <a href="chap7_mj.html">7</a>  <a href="chap8_mj.html">8</a>  <a href="chap9_mj.html">9</a>  <a href="chap10_mj.html">10</a>  <a href="chap11_mj.html">11</a>  <a href="chap12_mj.html">12</a>  <a href="chap13_mj.html">13</a>  <a href="chap14_mj.html">14</a>  <a href="chap15_mj.html">15</a>  <a href="chapBib_mj.html">Bib</a>  <a href="chapInd_mj.html">Ind</a>  </div>

<div class="chlinkprevnexttop"> <a href="chap0_mj.html">[Top of Book]</a>   <a href="chap0_mj.html#contents">[Contents]</a>    <a href="chap4_mj.html">[Previous Chapter]</a>    <a href="chap6_mj.html">[Next Chapter]</a>   </div>

<p id="mathjaxlink" class="pcenter"><a href="chap5.html">[MathJax off]</a></p>
<p><a id="X85CD9A43847AE1B8" name="X85CD9A43847AE1B8"></a></p>
<div class="ChapSects"><a href="chap5_mj.html#X85CD9A43847AE1B8">5 <span class="Heading">Whitehead group of a crossed module</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X7C01AE7783898705">5.1 <span class="Heading">Derivations and Sections</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X83EC6F7780F5636E">5.1-1 DerivationByImages</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7F119F9580C150B1">5.1-2 PrincipalDerivation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7E1C72897CD31D66">5.1-3 SectionByHomomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X87D9F7257DFF0236">5.1-4 IdentityDerivation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7AD6E23F8254F400">5.1-5 WhiteheadProduct</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X861A52407D3C627D">5.2 <span class="Heading">Whitehead Monoids and Groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X788884E48534F7CB">5.2-1 AllDerivations</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7CB1614E7EC58A84">5.2-2 WhiteheadMonoidTable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X84CD856C84BDB019">5.2-3 RegularDerivations</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X8452E45878691CD3">5.2-4 PrincipalDerivations</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X7E53AF1884B2D03D">5.3 <span class="Heading">Endomorphisms determined by a derivation</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X86AF32CA84217C46">5.3-1 SourceEndomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X84463DE2872AA709">5.3-2 RangeEndomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7E27AE6478566A94">5.3-3 Object2dEndomorphism</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X820501BF83D1D6D7">5.4 <span class="Heading">Whitehead groups for cat<span class="SimpleMath">\(^1\)</span>-groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X813CAA17855172E4">5.4-1 AllSections</a></span>
</div></div>
</div>

<h3>5 <span class="Heading">Whitehead group of a crossed module</span></h3>

<p><a id="X7C01AE7783898705" name="X7C01AE7783898705"></a></p>

<h4>5.1 <span class="Heading">Derivations and Sections</span></h4>

<p>The Whitehead monoid <span class="SimpleMath">\({\rm Der}(\calX)\)</span> of <span class="SimpleMath">\(\calX\)</span> was defined in <a href="chapBib_mj.html#biBW2">[Whi48]</a> to be the monoid of all <em>derivations</em> from <span class="SimpleMath">\(R\)</span> to <span class="SimpleMath">\(S\)</span>, that is the set of all maps <span class="SimpleMath">\(\chi : R \to S\)</span>, with <em>Whitehead product</em> <span class="SimpleMath">\(\star\)</span> (on the <em>right</em>) satisfying:</p>

<p class="center">\[
{\bf Der\ 1}:  \chi(qr) ~=~ (\chi q)^{r} \; (\chi r),
\qquad
{\bf Der\ 2}: (\chi_1 \star \chi_2)(r) 
    ~=~ (\chi_2 r)(\chi_1 r)(\chi_2 \partial \chi_1 r).
\]</p>

<p>It easily follows that <span class="SimpleMath">\(\chi 1 = 1\)</span> and <span class="SimpleMath">\(\chi(r^{-1}) = ((\chi r)^{-1})^{r^{-1}}\)</span>, and that the zero map is the identity for this composition. Invertible elements in the monoid are called <em>regular</em>. The Whitehead group of <span class="SimpleMath">\(\calX\)</span> is the group of regular derivations in <span class="SimpleMath">\({\rm Der}(\calX )\)</span>. In the next chapter the <em>actor</em> of <span class="SimpleMath">\(\calX\)</span> is defined as a crossed module whose source and range are permutation representations of the Whitehead group and the automorphism group of <span class="SimpleMath">\(\calX\)</span>.</p>

<p>The construction for cat<span class="SimpleMath">\(^1\)</span>-groups equivalent to the derivation of a crossed module is the <em>section</em>. The monoid of sections of <span class="SimpleMath">\(\calC = (e;t,h : G \to R)\)</span> is the set of group homomorphisms <span class="SimpleMath">\(\xi : R \to G\)</span>, with Whitehead multiplication <span class="SimpleMath">\(\star\)</span> (on the <em>right</em>) satisfying:</p>

<p class="center">\[
{\bf Sect\ 1}: t \circ \xi ~=~ {\rm id}_R,
\quad
{\bf Sect\ 2}: (\xi_1 \star \xi_2)(r) 
                ~=~ (\xi_1 r)(e h \xi_1 r)^{-1}(\xi_2 h \xi_1 r)  
                ~=~ (\xi_2 h \xi_1 r)(e h \xi_1 r)^{-1}(\xi_1 r).
\]</p>

<p>The embedding <span class="SimpleMath">\(e\)</span> is the identity for this composition, and <span class="SimpleMath">\(h(\xi_1 \star \xi_2) = (h \xi_1)(h \xi_2)\)</span>. A section is <em>regular</em> when <span class="SimpleMath">\(h \xi\)</span> is an automorphism, and the group of regular sections is isomorphic to the Whitehead group.</p>

<p>If <span class="SimpleMath">\(\epsilon\)</span> denotes the inclusion of <span class="SimpleMath">\(S = \ker\ t\)</span> in <span class="SimpleMath">\(G\)</span> then <span class="SimpleMath">\(\partial = h \epsilon : S \to R\)</span> and</p>

<p class="center">\[
\xi r ~=~ (e r)(\epsilon \chi r), 
\quad\mbox{which equals}\quad  
(r, \chi r) ~\in~ R \ltimes S, 
\]</p>

<p>determines a section <span class="SimpleMath">\(\xi\)</span> of <span class="SimpleMath">\(\calC\)</span> in terms of the corresponding derivation <span class="SimpleMath">\(\chi\)</span> of <span class="SimpleMath">\(\calX\)</span>, and conversely.</p>

<p><a id="X83EC6F7780F5636E" name="X83EC6F7780F5636E"></a></p>

<h5>5.1-1 DerivationByImages</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DerivationByImages</code>( <var class="Arg">X0</var>, <var class="Arg">ims</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">‣ IsDerivation</code>( <var class="Arg">map</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">‣ IsUp2DimensionalMapping</code>( <var class="Arg">chi</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">‣ UpGeneratorImages</code>( <var class="Arg">chi</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ UpImagePositions</code>( <var class="Arg">chi</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Object2d</code>( <var class="Arg">chi</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DerivationImage</code>( <var class="Arg">chi</var>, <var class="Arg">r</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>A derivation <span class="SimpleMath">\(\chi\)</span> is stored like a group homomorphisms by specifying the images of the generating set <code class="code">StrongGeneratorsStabChain( StabChain(R) )</code> of the range <span class="SimpleMath">\(R\)</span>. This set of images is stored as the attribute <code class="code">UpGeneratorImages</code> of <span class="SimpleMath">\(\chi\)</span>. The function <code class="code">IsDerivation</code> is automatically called to check that this procedure is well-defined. The attribute <code class="code">Object2d</code><span class="SimpleMath">\((\chi)\)</span> returns the underlying crossed module.</p>

<p>Images of the remaining elements may be obtained using axiom <span class="SimpleMath">\({\bf Der\ 1}\)</span>. <code class="code">UpImagePositions(chi)</code> is the list of the images under <span class="SimpleMath">\(\chi\)</span> of <code class="code">Elements(R)</code> and <code class="code">DerivationImage(chi,r)</code> returns <span class="SimpleMath">\(\chi r\)</span>.</p>

<p>In the following example a cat<span class="SimpleMath">\(^1\)</span>-group <code class="code">C3</code> and the associated crossed module <code class="code">X3</code> are constructed, where <code class="code">X3</code> is isomorphic to the inclusion of the normal cyclic group <code class="code">c3</code> in the symmetric group <code class="code">s3</code>. The derivation <span class="SimpleMath">\(\chi_1\)</span> maps <code class="code">c3</code> to the identity and the other <span class="SimpleMath">\(3\)</span> elements to <span class="SimpleMath">\((1,2,3)(4,6,5)\)</span>.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">g18 := Group( (1,2,3), (4,5,6), (2,3)(5,6) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetName( g18, "g18" );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">gen18 := GeneratorsOfGroup( g18 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">g1 := gen18[1];;  g2 := gen18[2];;  g3 := gen18[3];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">s3 := Subgroup( g18, gen18{[2..3]} );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetName( s3, "s3" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">t := GroupHomomorphismByImages( g18, s3, gen18, [g2,g2,g3] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">h := GroupHomomorphismByImages( g18, s3, gen18, [(),g2,g3] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">e := GroupHomomorphismByImages( s3, g18, [g2,g3], [g2,g3] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">C3 := Cat1Group( t, h, e );</span>
[g18=>s3]
<span class="GAPprompt">gap></span> <span class="GAPinput">SetName( Kernel(t), "c3" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">X3 := XModOfCat1Group( C3 );</span>
[c3->s3]
<span class="GAPprompt">gap></span> <span class="GAPinput">R3 := Range( X3 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">StrongGeneratorsStabChain( StabChain( R3 ) );</span>
[ (4,5,6), (2,3)(5,6) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">chi1 := DerivationByImages( X3, [ (), (1,2,3)(4,6,5) ] );</span>
DerivationByImages( s3, c3, [ (4,5,6), (2,3)(5,6) ], 
[ (), (1,2,3)(4,6,5) ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">[ IsUp2DimensionalMapping( chi1 ), IsDerivation( chi1 ) ];</span>
[ true, true ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Object2d( chi1 );</span>
[c3->s3]
<span class="GAPprompt">gap></span> <span class="GAPinput">UpGeneratorImages( chi1 ); </span>
[ (), (1,2,3)(4,6,5) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">UpImagePositions( chi1 );</span>
[ 1, 1, 1, 2, 2, 2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">DerivationImage( chi1, (2,3)(4,5) );</span>
(1,2,3)(4,6,5)

</pre></div>

<p><a id="X7F119F9580C150B1" name="X7F119F9580C150B1"></a></p>

<h5>5.1-2 PrincipalDerivation</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PrincipalDerivation</code>( <var class="Arg">X0</var>, <var class="Arg">s</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>The <em>principal derivation</em> determined by <span class="SimpleMath">\(s \in S\)</span> is the derivation <span class="SimpleMath">\(\eta_s : R \to S,\; r \mapsto (s^{-1})^rs\)</span>.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">eta := PrincipalDerivation( X3, (1,2,3)(4,6,5) );</span>
DerivationByImages( s3, c3, [ (4,5,6), (2,3)(5,6) ], [ (), (1,3,2)(4,5,6) ] )

</pre></div>

<p><a id="X7E1C72897CD31D66" name="X7E1C72897CD31D66"></a></p>

<h5>5.1-3 SectionByHomomorphism</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SectionByHomomorphism</code>( <var class="Arg">C</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">‣ IsSection</code>( <var class="Arg">xi</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">‣ UpHomomorphism</code>( <var class="Arg">xi</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SectionByDerivation</code>( <var class="Arg">chi</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">‣ DerivationBySection</code>( <var class="Arg">xi</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Sections <em>are</em> group homomorphisms but, although they do not require a special representation, one is provided in the category <code class="code">IsUp2DimensionalMapping</code> having attributes <code class="code">Object2d</code>, <code class="code">UpHomomorphism</code> and <code class="code">UpGeneratorsImages</code>. Operations <code class="code">SectionByDerivation</code> and <code class="code">DerivationBySection</code> convert derivations to sections, and vice-versa, calling <code class="func">Cat1GroupOfXMod</code> (<a href="chap2_mj.html#X82F10A59867C765D"><span class="RefLink">2.5-3</span></a>) and <code class="func">XModOfCat1Group</code> (<a href="chap2_mj.html#X82F10A59867C765D"><span class="RefLink">2.5-3</span></a>) automatically.</p>

<p>Two strategies for calculating derivations and sections are implemented, see <a href="chapBib_mj.html#biBAW1">[AW00]</a>. The default method for <code class="func">AllDerivations</code> (<a href="chap5_mj.html#X788884E48534F7CB"><span class="RefLink">5.2-1</span></a>) is to search for all possible sets of images using a backtracking procedure, and when all the derivations are found it is not known which are regular. In early versions of this package, the default method for <code class="code">AllSections( <C> )</code> was to compute all endomorphisms on the range group <code class="code">R</code> of <code class="code">C</code> as possibilities for the composite <span class="SimpleMath">\(h \xi\)</span>. A backtrack method then found possible images for such a section. In the current version the derivations of the associated crossed module are calculated, and these are all converted to sections using <code class="func">SectionByDerivation</code>.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">hom2 := GroupHomomorphismByImages( s3, g18, [ (4,5,6), (2,3)(5,6) ], </span>
<span class="GAPprompt">></span> <span class="GAPinput">[ (1,3,2)(4,6,5), (1,2)(4,6) ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">xi2 := SectionByHomomorphism( C3, hom2 );                                 </span>
SectionByHomomorphism( s3, g18, [ (4,5,6), (2,3)(5,6) ], 
[ (1,3,2)(4,6,5), (1,2)(4,6) ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">[ IsUp2DimensionalMapping( xi2 ), IsSection( xi2 ) ];</span>
[ true, true ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Object2d( xi2 );</span>
[g18 => s3]
<span class="GAPprompt">gap></span> <span class="GAPinput">UpHomomorphism( xi2 );         </span>
[ (4,5,6), (2,3)(5,6) ] -> [ (1,3,2)(4,6,5), (1,2)(4,6) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">UpGeneratorImages( xi2 );</span>
[ (1,3,2)(4,6,5), (1,2)(4,6) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">chi2 := DerivationBySection( xi2 );</span>
DerivationByImages( s3, c3, [ (4,5,6), (2,3)(5,6) ], 
[ (1,3,2)(4,5,6), (1,2,3)(4,6,5) ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">xi1 := SectionByDerivation( chi1 );</span>
SectionByHomomorphism( s3, g18, [ (4,5,6), (2,3)(5,6) ], 
[ (1,2,3), (1,2)(4,6) ] )

</pre></div>

<p><a id="X87D9F7257DFF0236" name="X87D9F7257DFF0236"></a></p>

<h5>5.1-4 IdentityDerivation</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IdentityDerivation</code>( <var class="Arg">X0</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IdentitySection</code>( <var class="Arg">C0</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The identity derivation maps the range group to the identity subgroup of the source, while the identity section is just the range embedding considered as a section.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">IdentityDerivation( X3 ); </span>
DerivationByImages( s3, c3, [ (4,5,6), (2,3)(5,6) ], [ (), () ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">IdentitySection( C3 );     </span>
SectionByHomomorphism( s3, g18, [ (4,5,6), (2,3)(5,6) ], 
[ (4,5,6), (2,3)(5,6) ] )

</pre></div>

<p><a id="X7AD6E23F8254F400" name="X7AD6E23F8254F400"></a></p>

<h5>5.1-5 WhiteheadProduct</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ WhiteheadProduct</code>( <var class="Arg">upi</var>, <var class="Arg">upj</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">‣ WhiteheadOrder</code>( <var class="Arg">up</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>The <code class="code">WhiteheadProduct</code>, defined in section <a href="chap5_mj.html#X7C01AE7783898705"><span class="RefLink">5.1</span></a>, may be applied to two derivations to form <span class="SimpleMath">\(\chi_i \star \chi_j\)</span>, or to two sections to form <span class="SimpleMath">\(\xi_i \star \xi_j\)</span>. The <code class="code">WhiteheadOrder</code> of a regular derivation <span class="SimpleMath">\(\chi\)</span> is the smallest power of <span class="SimpleMath">\(\chi\)</span>, using this product, equal to the <code class="func">IdentityDerivation</code> (<a href="chap5_mj.html#X87D9F7257DFF0236"><span class="RefLink">5.1-4</span></a>).</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">chi12 := WhiteheadProduct( chi1, chi2 );</span>
DerivationByImages( s3, c3, [ (4,5,6), (2,3)(5,6) ], [ (1,3,2)(4,5,6), () ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">xi12 := WhiteheadProduct( xi1, xi2 );</span>
SectionByHomomorphism( s3, g18, [ (4,5,6), (2,3)(5,6) ], 
[ (1,3,2)(4,6,5), (2,3)(5,6) ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">xi12 = SectionByDerivation( chi12 ); </span>
true 
<span class="GAPprompt">gap></span> <span class="GAPinput">[ WhiteheadOrder( chi2 ), WhiteheadOrder( xi2 ) ];</span>
[ 2, 2 ]

</pre></div>

<p><a id="X861A52407D3C627D" name="X861A52407D3C627D"></a></p>

<h4>5.2 <span class="Heading">Whitehead Monoids and Groups</span></h4>

<p>As mentioned at the beginning of this chapter, the Whitehead monoid <span class="SimpleMath">\({\rm Der}(\calX)\)</span> of <span class="SimpleMath">\(\calX\)</span> is the monoid of all derivations from <span class="SimpleMath">\(R\)</span> to <span class="SimpleMath">\(S\)</span>. Monoids of derivations have representation <code class="code">IsMonoidOfUp2DimensionalMappingsObj</code>. Multiplication tables for Whitehead monoids enable the construction of transformation representations.</p>

<p><a id="X788884E48534F7CB" name="X788884E48534F7CB"></a></p>

<h5>5.2-1 AllDerivations</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AllDerivations</code>( <var class="Arg">X0</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ImagesList</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DerivationClass</code>( <var class="Arg">mon</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ImagesTable</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Using our example <code class="code">X3</code> we find that there are just nine derivations. Here <code class="code">AllDerivations</code> returns a collection of up mappings with attributes:</p>


<ul>
<li><p><code class="code">Object2d</code> - the crossed modules <span class="SimpleMath">\(\calX\)</span>;</p>

</li>
<li><p><code class="code">ImagesList</code> - a list, for each derivation, of the images of the generators of the range group;</p>

</li>
<li><p><code class="code">DerivationClass</code> - the string "all"; other classes include "regular" and "principal";</p>

</li>
<li><p><code class="code">ImagesTable</code> - this is a table whose <span class="SimpleMath">\([i,j]\)</span>-th entry is the position in the list of elements of <span class="SimpleMath">\(S\)</span> of the image under the <span class="SimpleMath">\(i\)</span>-th derivation of the <span class="SimpleMath">\(j\)</span>-th element of <span class="SimpleMath">\(R\)</span>.</p>

</li>
</ul>

<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">all3 := AllDerivations( X3 );</span>
monoid of derivations with images list:
[ (), () ]
[ (), (1,3,2)(4,5,6) ]
[ (), (1,2,3)(4,6,5) ]
[ (1,3,2)(4,5,6), () ]
[ (1,3,2)(4,5,6), (1,3,2)(4,5,6) ]
[ (1,3,2)(4,5,6), (1,2,3)(4,6,5) ]
[ (1,2,3)(4,6,5), () ]
[ (1,2,3)(4,6,5), (1,3,2)(4,5,6) ]
[ (1,2,3)(4,6,5), (1,2,3)(4,6,5) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">DerivationClass( all3 );</span>
"all"
<span class="GAPprompt">gap></span> <span class="GAPinput">Perform( ImagesTable( all3 ), Display );</span>
[ 1, 1, 1, 1, 1, 1 ]
[ 1, 1, 1, 3, 3, 3 ]
[ 1, 1, 1, 2, 2, 2 ]
[ 1, 3, 2, 1, 3, 2 ]
[ 1, 3, 2, 3, 2, 1 ]
[ 1, 3, 2, 2, 1, 3 ]
[ 1, 2, 3, 1, 2, 3 ]
[ 1, 2, 3, 3, 1, 2 ]
[ 1, 2, 3, 2, 3, 1 ]

</pre></div>

<p><a id="X7CB1614E7EC58A84" name="X7CB1614E7EC58A84"></a></p>

<h5>5.2-2 WhiteheadMonoidTable</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ WhiteheadMonoidTable</code>( <var class="Arg">X0</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ WhiteheadTransformationMonoid</code>( <var class="Arg">X0</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The <code class="code">WhiteheadMonoidTable</code> of <span class="SimpleMath">\(\calX\)</span> is the multiplication table whose <span class="SimpleMath">\([i,j]\)</span>-th entry is the position <span class="SimpleMath">\(k\)</span> in the list of derivations of the Whitehead product <span class="SimpleMath">\(\chi_i*\chi_j = \chi_k\)</span>.</p>

<p>Using the rows of the table as transformations, we may construct the <code class="code">WhiteheadTransformationMonoid</code> of <span class="SimpleMath">\(\calX\)</span>.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">wmt3 := WhiteheadMonoidTable( X3 );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Perform( wmt3, Display );</span>
[ 1, 2, 3, 4, 5, 6, 7, 8, 9 ]
[ 2, 3, 1, 5, 6, 4, 8, 9, 7 ]
[ 3, 1, 2, 6, 4, 5, 9, 7, 8 ]
[ 4, 6, 5, 1, 3, 2, 7, 9, 8 ]
[ 5, 4, 6, 2, 1, 3, 8, 7, 9 ]
[ 6, 5, 4, 3, 2, 1, 9, 8, 7 ]
[ 7, 7, 7, 7, 7, 7, 7, 7, 7 ]
[ 8, 8, 8, 8, 8, 8, 8, 8, 8 ]
[ 9, 9, 9, 9, 9, 9, 9, 9, 9 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">wtm3 := WhiteheadTransformationMonoid( X3 );</span>
<transformation monoid of degree 9 with 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfMonoid( wtm3 ); </span>
[ Transformation( [ 2, 3, 1, 5, 6, 4, 8, 9, 7 ] ), 
  Transformation( [ 4, 6, 5, 1, 3, 2, 7, 9, 8 ] ), 
  Transformation( [ 7, 7, 7, 7, 7, 7, 7, 7, 7 ] ) ]

</pre></div>

<p><a id="X84CD856C84BDB019" name="X84CD856C84BDB019"></a></p>

<h5>5.2-3 RegularDerivations</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RegularDerivations</code>( <var class="Arg">X0</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ WhiteheadGroupTable</code>( <var class="Arg">X0</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ WhiteheadPermGroup</code>( <var class="Arg">X0</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsWhiteheadPermGroup</code>( <var class="Arg">G</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">‣ WhiteheadRegularGroup</code>( <var class="Arg">X0</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ WhiteheadGroupIsomorphism</code>( <var class="Arg">X0</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p><code class="code">RegularDerivations</code> are those derivations which are invertible in the monoid. The multiplication table for a Whitehead group - a subtable of the Whitehead monoid table - enables the construction of a permutation representation <code class="code">WhiteheadPermGroup</code> <span class="SimpleMath">\(W\calX\)</span> of <span class="SimpleMath">\(\calX\)</span>. This group satisfies the property <code class="code">IsWhiteheadPermGroup</code>. and <span class="SimpleMath">\(\calX\)</span> is its attribute <code class="code">Object2d</code>.</p>

<p>Of the nine derivations of <code class="code">X3</code> just six are regular. The associated group is isomorphic to the symmetric group <code class="code">s3</code>.</p>

<p>Taking the rows of the <code class="code">WhiteheadGroupTable</code> as permutations, we may construct the <code class="code">WhiteheadRegularGroup</code> of <span class="SimpleMath">\(\calX\)</span>. Then, seeking a <code class="code">SmallerDegreePermutationRepresentation</code>, we obtain the <code class="code">WhiteheadGroupIsomorphism</code> whose image is the <code class="code">WhiteheadPermGroup</code> of <span class="SimpleMath">\(\calX\)</span>.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">reg3 := RegularDerivations( X3 );</span>
monoid of derivations with images list:
[ (), () ]
[ (), (1,3,2)(4,5,6) ]
[ (), (1,2,3)(4,6,5) ]
[ (1,3,2)(4,5,6), () ]
[ (1,3,2)(4,5,6), (1,3,2)(4,5,6) ]
[ (1,3,2)(4,5,6), (1,2,3)(4,6,5) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">DerivationClass( reg3 );</span>
"regular"
<span class="GAPprompt">gap></span> <span class="GAPinput">wgt3 := WhiteheadGroupTable( X3 );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Perform( wgt3, Display );</span>
[ [ 1, 2, 3, 4, 5, 6 ],
  [ 2, 3, 1, 5, 6, 4 ],
  [ 3, 1, 2, 6, 4, 5 ],
  [ 4, 6, 5, 1, 3, 2 ],
  [ 5, 4, 6, 2, 1, 3 ],
  [ 6, 5, 4, 3, 2, 1 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">wpg3 := WhiteheadPermGroup( X3 );</span>
Group([ (1,2,3), (1,2) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">IsWhiteheadPermGroup( wpg3 );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">Object2d( wpg3 );</span>
[c3->s3]
<span class="GAPprompt">gap></span> <span class="GAPinput">WhiteheadRegularGroup( X3 );</span>
Group([ (1,2,3)(4,5,6), (1,4)(2,6)(3,5) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">MappingGeneratorsImages( WhiteheadGroupIsomorphism( X3 ) );</span>
[ [ (1,2,3)(4,5,6), (1,4)(2,6)(3,5) ], [ (1,2,3), (1,2) ] ]

</pre></div>

<p><a id="X8452E45878691CD3" name="X8452E45878691CD3"></a></p>

<h5>5.2-4 PrincipalDerivations</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PrincipalDerivations</code>( <var class="Arg">X0</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PrincipalDerivationSubgroup</code>( <var class="Arg">X0</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ WhiteheadHomomorphism</code>( <var class="Arg">X0</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The principal derivations form a subgroup of the Whitehead group. The <code class="code">PrincipalDerivationSubgroup</code> is the corresponding subgroup of the <code class="code">WhiteheadPermGroup</code>.</p>

<p>The Whitehead homomorphism <span class="SimpleMath">\(\eta : S \to W\calX, s \mapsto \eta_s\)</span> for <span class="SimpleMath">\(\calX\)</span> maps the source group of <span class="SimpleMath">\(\calX\)</span> to the Whitehead group of <span class="SimpleMath">\(\calX\)</span>.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">PDX3 := PrincipalDerivations( X3 );</span>
monoid of derivations with images list:
[ (), () ]
[ (), (1,3,2)(4,5,6) ]
[ (), (1,2,3)(4,6,5) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">PDSX3 := PrincipalDerivationSubgroup( X3 );</span>
Group([ (1,2,3) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">Whom3 := WhiteheadHomomorphism( X3 );</span>
[ (1,2,3)(4,6,5) ] -> [ (1,2,3) ]

</pre></div>

<p><strong class="button">Exercise:</strong><span class="SimpleMath">\(~\)</span> Use the two crossed module axioms to show that <span class="SimpleMath">\(\eta_{s_1} \star \eta_{s_2} = \eta_{s_1s_2}\)</span>.</p>

<p><a id="X7E53AF1884B2D03D" name="X7E53AF1884B2D03D"></a></p>

<h4>5.3 <span class="Heading">Endomorphisms determined by a derivation</span></h4>

<p><a id="X86AF32CA84217C46" name="X86AF32CA84217C46"></a></p>

<h5>5.3-1 SourceEndomorphism</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SourceEndomorphism</code>( <var class="Arg">chi</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>A derivation <span class="SimpleMath">\(\chi\)</span> of <span class="SimpleMath">\(\calX = (\partial : S \to R)\)</span> determines an endomorphism <span class="SimpleMath">\(\sigma_{\chi} : S \to S,~ s \mapsto s(\chi \partial s)\)</span>. We may verify that <span class="SimpleMath">\(\sigma_{\chi}\)</span> is a homomorphism by:</p>

<p class="center">\[
\sigma_{\chi}(s_1s_2) 
~=~ s_1s_2\chi((\partial s_1)(\partial s_2))
~=~ s_1s_2(\chi\partial s_1)^{\partial s_2}(\chi\partial s_2)
~=~ s_1(\chi\partial s_1)s_2(\chi\partial s_2)
~=~ (\sigma_{\chi}s_1)(\sigma_{\chi}s_2).
\]</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">sigma2 := SourceEndomorphism( chi2 );</span>
[ (1,2,3)(4,6,5) ] -> [ (1,3,2)(4,5,6) ]

</pre></div>

<p><a id="X84463DE2872AA709" name="X84463DE2872AA709"></a></p>

<h5>5.3-2 RangeEndomorphism</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RangeEndomorphism</code>( <var class="Arg">chi</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>A derivation <span class="SimpleMath">\(\chi\)</span> of <span class="SimpleMath">\(\calX = (\partial : S \to R)\)</span> determines an endomorphism <span class="SimpleMath">\(\rho_{\chi} : R \to R,~ r \mapsto r(\partial \chi r)\)</span>.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">rho2 := RangeEndomorphism( chi2 );</span>
[ (4,5,6), (2,3)(5,6) ] -> [ (4,6,5), (2,3)(4,6) ]

</pre></div>

<p><a id="X7E27AE6478566A94" name="X7E27AE6478566A94"></a></p>

<h5>5.3-3 Object2dEndomorphism</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Object2dEndomorphism</code>( <var class="Arg">chi</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>A derivation <span class="SimpleMath">\(\chi\)</span> of <span class="SimpleMath">\(\calX = (\partial : S \to R)\)</span> determines an endomorphism <span class="SimpleMath">\(\alpha_{\chi} : \calX \to \calX\)</span> whose source and range endomorphisms are given by the previous two operations.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">alpha2 := Object2dEndomorphism( chi2 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( alpha2 );</span>
Morphism of crossed modules :- 
: Source = [c3->s3] with generating sets:
  [ (1,2,3)(4,6,5) ]
  [ (4,5,6), (2,3)(5,6) ]
: Range = Source
: Source Homomorphism maps source generators to:
  [ (1,3,2)(4,5,6) ]
: Range Homomorphism maps range generators to:
  [ (4,6,5), (2,3)(4,6) ]

</pre></div>

<p><a id="X820501BF83D1D6D7" name="X820501BF83D1D6D7"></a></p>

<h4>5.4 <span class="Heading">Whitehead groups for cat<span class="SimpleMath">\(^1\)</span>-groups</span></h4>

<p><a id="X813CAA17855172E4" name="X813CAA17855172E4"></a></p>

<h5>5.4-1 AllSections</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AllSections</code>( <var class="Arg">C0</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RegularSections</code>( <var class="Arg">C0</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>These operations are currently obtained by running the equivalent operation for derivations and then converting the result to sections.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">AllSections( C3 );</span>
monoid of sections with images list:
[ (4,5,6), (2,3)(5,6) ]
[ (4,5,6), (1,3)(4,5) ]
[ (4,5,6), (1,2)(4,6) ]
[ (1,3,2)(4,6,5), (2,3)(5,6) ]
[ (1,3,2)(4,6,5), (1,3)(4,5) ]
[ (1,3,2)(4,6,5), (1,2)(4,6) ]
[ (1,2,3), (2,3)(5,6) ]
[ (1,2,3), (1,3)(4,5) ]
[ (1,2,3), (1,2)(4,6) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">RegularSections( C3 );         </span>
monoid of sections with images list:
[ (4,5,6), (2,3)(5,6) ]
[ (4,5,6), (1,3)(4,5) ]
[ (4,5,6), (1,2)(4,6) ]
[ (1,3,2)(4,6,5), (2,3)(5,6) ]
[ (1,3,2)(4,6,5), (1,3)(4,5) ]
[ (1,3,2)(4,6,5), (1,2)(4,6) ]

</pre></div>


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