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<p id="mathjaxlink" class="pcenter"><a href="chap9_mj.html">[MathJax on]</a></p>
<p><a id="X7B8C20A9826087E1" name="X7B8C20A9826087E1"></a></p>
<div class="ChapSects"><a href="chap9.html#X7B8C20A9826087E1">9 <span class="Heading">Group derivations</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9.html#X7AAB25B587D3DF70">9.1 <span class="Heading">Creating group derivations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X8303ADE37FFAA109">9.1-1 GroupDerivationByImages</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X7C9D096A7B996E89">9.1-2 GroupDerivationByFunction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X8341EA2B7FBAE696">9.1-3 GroupDerivationByAffineAction</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9.html#X7AE626D685C68CF0">9.2 <span class="Heading">Operations for group derivations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X7F065FD7822C0A12">9.2-1 <span class="Heading">IsInjective</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X784ECE847E005B8F">9.2-2 <span class="Heading">IsSurjective</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X878F56AB7B342767">9.2-3 <span class="Heading">IsBijective</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X7DCD99628504B810">9.2-4 <span class="Heading">Kernel</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X87F4D35A826599C6">9.2-5 <span class="Heading">Image</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X7AE24A1586B7DE79">9.2-6 <span class="Heading">PreImagesRepresentative</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X85C8590E832002EF">9.2-7 <span class="Heading">PreImages</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9.html#X801FDEFE8155D0B1">9.3 <span class="Heading">Images of group derivations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X87BDB89B7AAFE8AD">9.3-1 <span class="Heading">\in</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X858ADA3B7A684421">9.3-2 <span class="Heading">Size</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X7EBA57FC7CCF8449">9.3-3 <span class="Heading">List</span></a>
</span>
</div></div>
</div>

<h3>9 <span class="Heading">Group derivations</span></h3>

<p>Let <span class="Math">G</span> and <span class="Math">H</span> be groups and let <span class="Math">H</span> act on <span class="Math">G</span> via automorphisms, i.e. there is a group homomorphism</p>

<p class="pcenter">\alpha \colon H \to \operatorname{Aut}(G) \colon h \mapsto \alpha_h</p>

<p>such that <span class="Math">g^h = \alpha_h(g)</span> for all <span class="Math">g \in G</span> and <span class="Math">h \in H</span>. A <em>group derivation</em> <span class="Math">\delta \colon H \to G</span> is a map such that</p>

<p class="pcenter">\delta(h_1h_2) = \delta(h_1)^{h_2}\delta(h_2).</p>

<p>Note that we do not require <span class="Math">G</span> to be abelian.</p>

<p>Algorithms designed for computing with twisted conjugacy classes can be leveraged to do computations involving group derivations, see <a href="chapBib.html#biBtert25-a">[Ter25, Sec. 10]</a> for a description on this.</p>

<p>Please note that the functions in this chapter require <span class="Math">G</span> and <span class="Math">H</span> to either both be finite, or both be PcpGroups.</p>

<p><a id="X7AAB25B587D3DF70" name="X7AAB25B587D3DF70"></a></p>

<h4>9.1 <span class="Heading">Creating group derivations</span></h4>

<p><a id="X8303ADE37FFAA109" name="X8303ADE37FFAA109"></a></p>

<h5>9.1-1 GroupDerivationByImages</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GroupDerivationByImages</code>( <var class="Arg">H</var>, <var class="Arg">G</var>[[, <var class="Arg">gens</var>], <var class="Arg">imgs</var>], <var class="Arg">act</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">‣ GroupDerivationByImagesNC</code>( <var class="Arg">H</var>, <var class="Arg">G</var>[[, <var class="Arg">gens</var>], <var class="Arg">imgs</var>], <var class="Arg">act</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: the specified group derivation, or <code class="keyw">fail</code> if the given arguments do not define a derivation.</p>

<p>This works in the same vein as <code class="func">GroupHomomorphismByImages</code> (<a href="/home/runner/gap/doc/ref/chap40.html#X7F348F497C813BE0"><span class="RefLink">Ref 40.1-1</span></a>). The group <var class="Arg">H</var> acts on the group <var class="Arg">G</var> via <var class="Arg">act</var>, which must be a homomorphism from <var class="Arg">H</var> into a group of automorphisms of <var class="Arg">G</var>. This command then returns the group derivation defined by mapping the list <var class="Arg">gens</var> of generators of <var class="Arg">H</var> to the list <var class="Arg">imgs</var> of images in <var class="Arg">G</var>.</p>

<p>If omitted, the arguments <var class="Arg">gens</var> and <var class="Arg">imgs</var> default to the <code class="code">GeneratorsOfGroup</code> value of <var class="Arg">H</var> and <var class="Arg">G</var> respectively.</p>

<p>This function checks whether <var class="Arg">gens</var> generate <var class="Arg">H</var> and whether the mapping of the generators extends to a group derivation. This test can be expensive, so if one is certain that the given arguments produce a group derivation, these checks can be avoided by using the <code class="code">NC</code> version.</p>

<p><a id="X7C9D096A7B996E89" name="X7C9D096A7B996E89"></a></p>

<h5>9.1-2 GroupDerivationByFunction</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GroupDerivationByFunction</code>( <var class="Arg">H</var>, <var class="Arg">G</var>, <var class="Arg">fun</var>, <var class="Arg">act</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: the specified group derivation.</p>

<p><code class="code">GroupDerivationByFunction</code> works in the same vein as <code class="func">GroupHomomorphismByFunction</code> (<a href="/home/runner/gap/doc/ref/chap40.html#X7BC6C20E7CEDBFC5"><span class="RefLink">Ref 40.1-4</span></a>). The group <var class="Arg">H</var> acts on the group <var class="Arg">G</var> via <var class="Arg">act</var>, which must be a homomorphism from <var class="Arg">H</var> into a group of automorphisms of <var class="Arg">G</var>. This command then returns the group derivation defined by mapping the element <code class="code">h</code> of <var class="Arg">H</var> to the element <var class="Arg">fun</var>( <code class="code">h</code> ) of <var class="Arg">G</var>, where <var class="Arg">fun</var> is a <strong class="pkg">GAP</strong> function.</p>

<p>No tests are performed to check whether the arguments really produce a group derivation.</p>

<p><a id="X8341EA2B7FBAE696" name="X8341EA2B7FBAE696"></a></p>

<h5>9.1-3 GroupDerivationByAffineAction</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GroupDerivationByAffineAction</code>( <var class="Arg">H</var>, <var class="Arg">G</var>, <var class="Arg">act</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: the derivation that makes up the translational part of the affine action.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">H := PcGroupCode( 149167619499417164, 72 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := PcGroupCode( 5551210572, 72 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">inn := InnerAutomorphism( G, G.2 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">hom := GroupHomomorphismByImages(</span>
<span class="GAPprompt">></span> <span class="GAPinput">     G, G,</span>
<span class="GAPprompt">></span> <span class="GAPinput">     [ G.1*G.2, G.5 ], [ G.1*G.2^2*G.3^2*G.4, G.5 ]</span>
<span class="GAPprompt">></span> <span class="GAPinput">   );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">act := GroupHomomorphismByImages(</span>
<span class="GAPprompt">></span> <span class="GAPinput">     H, AutomorphismGroup( G ),</span>
<span class="GAPprompt">></span> <span class="GAPinput">     [ H.2, H.1*H.4 ], [ inn, hom ]</span>
<span class="GAPprompt">></span> <span class="GAPinput">   );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">gens := [ H.2, H.1*H.4 ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">imgs := [ G.5, G.2 ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">der := GroupDerivationByImages( H, G, gens, imgs, act );</span>
Group derivation [ f2, f1*f4 ] -> [ f5, f2 ]
</pre></div>

<p><a id="X7AE626D685C68CF0" name="X7AE626D685C68CF0"></a></p>

<h4>9.2 <span class="Heading">Operations for group derivations</span></h4>

<p>Many of the functions, operations, attributes... available to group homomorphisms are available for group derivations as well. We list some of the more useful ones.</p>

<p><a id="X7F065FD7822C0A12" name="X7F065FD7822C0A12"></a></p>

<h5>9.2-1 <span class="Heading">IsInjective</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsInjective</code>( <var class="Arg">der</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> if the group derivation <var class="Arg">der</var> is injective, otherwise <code class="keyw">false</code>.</p>

<p><a id="X784ECE847E005B8F" name="X784ECE847E005B8F"></a></p>

<h5>9.2-2 <span class="Heading">IsSurjective</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSurjective</code>( <var class="Arg">der</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> if the group derivation <var class="Arg">der</var> is surjective, otherwise <code class="keyw">false</code>.</p>

<p><a id="X878F56AB7B342767" name="X878F56AB7B342767"></a></p>

<h5>9.2-3 <span class="Heading">IsBijective</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsBijective</code>( <var class="Arg">der</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> if the group derivation <var class="Arg">der</var> is bijjective, otherwise <code class="keyw">false</code>.</p>

<p><a id="X7DCD99628504B810" name="X7DCD99628504B810"></a></p>

<h5>9.2-4 <span class="Heading">Kernel</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Kernel</code>( <var class="Arg">der</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: the set of elements that are mapped to the identity by <var class="Arg">der</var>.</p>

<p>This will always be a subgroup of <code class="code">Source</code>(<var class="Arg">der</var>).</p>

<p><a id="X87F4D35A826599C6" name="X87F4D35A826599C6"></a></p>

<h5>9.2-5 <span class="Heading">Image</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Image</code>( <var class="Arg">der</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">‣ Image</code>( <var class="Arg">der</var>, <var class="Arg">elm</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">‣ Image</code>( <var class="Arg">der</var>, <var class="Arg">sub</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: the image of the group derivation <var class="Arg">der</var>.</p>

<p>One can optionally give an element <var class="Arg">elm</var> or a subgroup <var class="Arg">sub</var> as a second argument, in which case <code class="code">Image</code> will calculate the image of this argument under <var class="Arg">der</var>.</p>

<p><a id="X7AE24A1586B7DE79" name="X7AE24A1586B7DE79"></a></p>

<h5>9.2-6 <span class="Heading">PreImagesRepresentative</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PreImagesRepresentative</code>( <var class="Arg">der</var>, <var class="Arg">elm</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a preimage of the element <var class="Arg">elm</var> under the group derivation <var class="Arg">der</var>, or <code class="keyw">fail</code> if no preimage exists.</p>

<p><a id="X85C8590E832002EF" name="X85C8590E832002EF"></a></p>

<h5>9.2-7 <span class="Heading">PreImages</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PreImages</code>( <var class="Arg">der</var>, <var class="Arg">elm</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: the set of all preimages of the element <var class="Arg">elm</var> under the group derivation <var class="Arg">der</var>.</p>

<p>This will always be a (right) coset of <code class="code">Kernel</code>( <var class="Arg">der</var> ), or the empty list.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsInjective( der ) or IsSurjective( der );</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">K := Kernel( der );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size( K );</span>
9
<span class="GAPprompt">gap></span> <span class="GAPinput">ImH := Image( der );</span>
Group derivation image in Group( [ f1, f2, f3, f4, f5 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">h1 := H.1*H.3;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">g := Image( der, h1 );</span>
f2*f4
<span class="GAPprompt">gap></span> <span class="GAPinput">ImK := Image( der, K );</span>
Group derivation image in Group( [ f1, f2, f3, f4, f5 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">h2 := PreImagesRepresentative( der, g );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Image( der, h2 ) = g;</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">PreIm := PreImages( der, g );</span>
RightCoset(<group of size 9 with 2 generators>,<object>)
<span class="GAPprompt">gap></span> <span class="GAPinput">PreIm = RightCoset( K, h2 );</span>
true
</pre></div>

<p><a id="X801FDEFE8155D0B1" name="X801FDEFE8155D0B1"></a></p>

<h4>9.3 <span class="Heading">Images of group derivations</span></h4>

<p>In general, the image of a group derivation is not a subgroup. However, it is still possible to do a membership test, to calculate the number of elements, and to enumerate the elements if there are only finitely many.</p>

<p><a id="X87BDB89B7AAFE8AD" name="X87BDB89B7AAFE8AD"></a></p>

<h5>9.3-1 <span class="Heading">\in</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ \in</code>( <var class="Arg">elm</var>, <var class="Arg">img</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> if <var class="Arg">elm</var> is an element of <var class="Arg">img</var>, otherwise <code class="keyw">false</code>.</p>

<p><a id="X858ADA3B7A684421" name="X858ADA3B7A684421"></a></p>

<h5>9.3-2 <span class="Heading">Size</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Size</code>( <var class="Arg">img</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: the number of elements in <var class="Arg">img</var>.</p>

<p><a id="X7EBA57FC7CCF8449" name="X7EBA57FC7CCF8449"></a></p>

<h5>9.3-3 <span class="Heading">List</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ List</code>( <var class="Arg">img</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a list containing the elements of <var class="Arg">img</var>.</p>

<p>If <var class="Arg">img</var> is infinite, this will run forever. It is recommended to first test the finiteness of <var class="Arg">img</var> using <code class="func">Size</code> (<a href="chap9.html#X858ADA3B7A684421"><span class="RefLink">9.3-2</span></a>).</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size( ImH );</span>
8
<span class="GAPprompt">gap></span> <span class="GAPinput">Size( ImK );</span>
1
<span class="GAPprompt">gap></span> <span class="GAPinput">g in ImH;</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">g in ImK;</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">List( ImK );</span>
[ <identity> of ... ]
</pre></div>


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