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<p id="mathjaxlink" class="pcenter"><a href="chap10_mj.html">[MathJax on]</a></p>
<p><a id="X87A5683C7B645EA1" name="X87A5683C7B645EA1"></a></p>
<div class="ChapSects"><a href="chap10.html#X87A5683C7B645EA1">10 <span class="Heading">Affine actions</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap10.html#X7E00F3E17A88ED4B">10.1 <span class="Heading">Creating an affine action</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X8116545B7DBE00AC">10.1-1 AffineActionByGroupDerivation</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap10.html#X86DD85AA827068A2">10.2 <span class="Heading">Operations for affine actions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X84AFABF98784C123">10.2-1 <span class="Heading">OrbitAffineAction</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X7E8F571A83D951B0">10.2-2 <span class="Heading">OrbitsAffineAction</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X8020B50487227359">10.2-3 <span class="Heading">NrOrbitsAffineAction</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X860FBE2378E0696D">10.2-4 <span class="Heading">StabiliserAffineAction</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X7B111FAB7D2A8C99">10.2-5 <span class="Heading">RepresentativeAffineAction</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap10.html#X81B54C657AE4B06F">10.3 <span class="Heading">Operations on orbits of affine actions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X865507568182424E">10.3-1 <span class="Heading">Representative</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X7B9DB15D80CE28B4">10.3-2 <span class="Heading">ActingDomain</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X86153CB087394DC1">10.3-3 <span class="Heading">FunctionAction</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X87BDB89B7AAFE8AD">10.3-4 <span class="Heading">\in</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X858ADA3B7A684421">10.3-5 <span class="Heading">Size</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X867840C67C990840">10.3-6 <span class="Heading">StabiliserOfExternalSet</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X7EBA57FC7CCF8449">10.3-7 <span class="Heading">List</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X79730D657AB219DB">10.3-8 <span class="Heading">Random</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X806A4814806A4814">10.3-9 <span class="Heading">\=</span></a>
</span>
</div></div>
</div>

<h3>10 <span class="Heading">Affine actions</span></h3>

<p>Let <span class="Math">G</span> and <span class="Math">H</span> be groups, let <span class="Math">H</span> act on <span class="Math">G</span> (via automorphisms) by</p>

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

<p>and let <span class="Math">\delta \colon H \to G</span> be a group derivation with respect to this action. Then we can construct a new action, called the <em>affine action</em> associated to <span class="Math">\delta</span>, by</p>

<p class="pcenter">G \times H \to G \colon g^h = \alpha_h(g) \delta(h).</p>

<p>If <span class="Math">K</span> is a subgroup of <span class="Math">H</span>, then the restriction of the affine action of <span class="Math">H</span> on <span class="Math">G</span> to <span class="Math">K</span> coincides with the affine action of <span class="Math">K</span> on <span class="Math">G</span> associated to the restriction of <span class="Math">\delta</span> to <span class="Math">K</span>.</p>

<p>Algorithms designed for computing with twisted conjugacy classes can be leveraged to do computations involving affine actions, 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="X7E00F3E17A88ED4B" name="X7E00F3E17A88ED4B"></a></p>

<h4>10.1 <span class="Heading">Creating an affine action</span></h4>

<p><a id="X8116545B7DBE00AC" name="X8116545B7DBE00AC"></a></p>

<h5>10.1-1 AffineActionByGroupDerivation</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AffineActionByGroupDerivation</code>( <var class="Arg">K</var>, <var class="Arg">der</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: the affine action of <var class="Arg">K</var> associated to the derivation <var class="Arg">der</var>.</p>

<p>The group <var class="Arg">K</var> must be a subgroup of <code class="code">Source(<var class="Arg">der</var>)</code>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">aff := AffineActionByGroupDerivation( H, der );</span>
function( g, k ) ... end
</pre></div>

<p><a id="X86DD85AA827068A2" name="X86DD85AA827068A2"></a></p>

<h4>10.2 <span class="Heading">Operations for affine actions</span></h4>

<p>These functions are analogues of existing <strong class="pkg">GAP</strong> functions for group actions.</p>

<p><a id="X84AFABF98784C123" name="X84AFABF98784C123"></a></p>

<h5>10.2-1 <span class="Heading">OrbitAffineAction</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OrbitAffineAction</code>( <var class="Arg">K</var>, <var class="Arg">g</var>, <var class="Arg">der</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: the orbit of <var class="Arg">g</var> under the affine action of <var class="Arg">K</var> associated to <var class="Arg">der</var>.</p>

<p>The group <var class="Arg">K</var> must be a subgroup of <code class="code">Source(<var class="Arg">der</var>)</code>.</p>

<p><a id="X7E8F571A83D951B0" name="X7E8F571A83D951B0"></a></p>

<h5>10.2-2 <span class="Heading">OrbitsAffineAction</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OrbitsAffineAction</code>( <var class="Arg">K</var>, <var class="Arg">der</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a list containing the orbits under the affine action of <var class="Arg">K</var> associated to <var class="Arg">der</var> if there are finitely many, or <code class="keyw">fail</code> if there are infinitely many.</p>

<p>The group <var class="Arg">K</var> must be a subgroup of <code class="code">Source(<var class="Arg">der</var>)</code>.</p>

<p><a id="X8020B50487227359" name="X8020B50487227359"></a></p>

<h5>10.2-3 <span class="Heading">NrOrbitsAffineAction</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NrOrbitsAffineAction</code>( <var class="Arg">K</var>, <var class="Arg">der</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: the number of orbits under the affine action of <var class="Arg">K</var> associated to <var class="Arg">der</var>.</p>

<p><a id="X860FBE2378E0696D" name="X860FBE2378E0696D"></a></p>

<h5>10.2-4 <span class="Heading">StabiliserAffineAction</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ StabiliserAffineAction</code>( <var class="Arg">K</var>, <var class="Arg">g</var>, <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">‣ StabilizerAffineAction</code>( <var class="Arg">K</var>, <var class="Arg">g</var>, <var class="Arg">der</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: the stabiliser of <var class="Arg">g</var> under the affine action of <var class="Arg">K</var> associated to <var class="Arg">der</var>.</p>

<p>The group <var class="Arg">K</var> must be a subgroup of <code class="code">Source(<var class="Arg">der</var>)</code>.</p>

<p><a id="X7B111FAB7D2A8C99" name="X7B111FAB7D2A8C99"></a></p>

<h5>10.2-5 <span class="Heading">RepresentativeAffineAction</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RepresentativeAffineAction</code>( <var class="Arg">K</var>, <var class="Arg">g1</var>, <var class="Arg">g2</var>, <var class="Arg">der</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: an element of <var class="Arg">K</var> that maps <var class="Arg">g1</var> to <var class="Arg">g2</var> under the affine action of <var class="Arg">K</var> associated to <var class="Arg">der</var>, or <code class="keyw">fail</code> if no such element exists.</p>

<p>The group <var class="Arg">K</var> must be a subgroup of <code class="code">Source(<var class="Arg">der</var>)</code>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g1 := G.1;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">orb := OrbitAffineAction( H, g1, der );</span>
f1^G
<span class="GAPprompt">gap></span> <span class="GAPinput">NrOrbitsAffineAction( H, der );</span>
10
<span class="GAPprompt">gap></span> <span class="GAPinput">stab := StabiliserAffineAction( H, g1, der );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Set( stab );</span>
[ <identity> of ..., f3, f3^2, f2^2*f5, f2*f4*f5,
  f2^2*f3*f5, f2*f3*f4*f5, f2^2*f3^2*f5, f2*f3^2*f4*f5 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">g2 := G.1*G.4*G.5;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">h := RepresentativeAffineAction( H, g1, g2, der );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">aff( g1, h ) = g2;</span>
true
</pre></div>

<p><a id="X81B54C657AE4B06F" name="X81B54C657AE4B06F"></a></p>

<h4>10.3 <span class="Heading">Operations on orbits of affine actions</span></h4>

<p><a id="X865507568182424E" name="X865507568182424E"></a></p>

<h5>10.3-1 <span class="Heading">Representative</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Representative</code>( <var class="Arg">orb</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: the group element that was used to construct <var class="Arg">orb</var>.</p>

<p><a id="X7B9DB15D80CE28B4" name="X7B9DB15D80CE28B4"></a></p>

<h5>10.3-2 <span class="Heading">ActingDomain</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ActingDomain</code>( <var class="Arg">orb</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: the group whose affine action <var class="Arg">orb</var> is an orbit of.</p>

<p><a id="X86153CB087394DC1" name="X86153CB087394DC1"></a></p>

<h5>10.3-3 <span class="Heading">FunctionAction</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FunctionAction</code>( <var class="Arg">orb</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: the affine action that <var class="Arg">orb</var> is an orbit of.</p>

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

<h5>10.3-4 <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">orb</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">orb</var>, otherwise <code class="keyw">false</code>.</p>

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

<h5>10.3-5 <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">orb</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: the number of elements in <var class="Arg">orb</var>.</p>

<p><a id="X867840C67C990840" name="X867840C67C990840"></a></p>

<h5>10.3-6 <span class="Heading">StabiliserOfExternalSet</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ StabiliserOfExternalSet</code>( <var class="Arg">orb</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: the stabiliser of <code class="code">Representative(<var class="Arg">orb</var>)</code> under the action <code class="code">FunctionAction(<var class="Arg">orb</var>)</code>.</p>

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

<h5>10.3-7 <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">orb</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a list containing the elements of <var class="Arg">orb</var>.</p>

<p>If <var class="Arg">orb</var> is infinite, this will run forever. It is recommended to first test the finiteness of <var class="Arg">orb</var> using <code class="func">Size</code> (<a href="chap10.html#X858ADA3B7A684421"><span class="RefLink">10.3-5</span></a>).</p>

<p><a id="X79730D657AB219DB" name="X79730D657AB219DB"></a></p>

<h5>10.3-8 <span class="Heading">Random</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Random</code>( <var class="Arg">orb</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a random element in <var class="Arg">orb</var>.</p>

<p><a id="X806A4814806A4814" name="X806A4814806A4814"></a></p>

<h5>10.3-9 <span class="Heading">\=</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ \=</code>( <var class="Arg">orb1</var>, <var class="Arg">orb2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> if <var class="Arg">orb1</var> is equal to <var class="Arg">orb2</var>, otherwise <code class="keyw">false</code>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g2 in orb;</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">G.2 in orb;</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">Size( orb );</span>
8
</pre></div>


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