<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>
<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>
<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>
<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>
<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="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>
<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>
<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>
<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>
¤ Die Informationen auf dieser Webseite wurden
nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit,
noch Qualität der bereit gestellten Informationen zugesichert.0.20Bemerkung:
Wie Sie bei der Firma Beratungs- und Dienstleistungen beauftragen können
¤
Die Informationen auf dieser Webseite wurden
nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit,
noch Qualität der bereit gestellten Informationen zugesichert.
Bemerkung:
Die farbliche Syntaxdarstellung ist noch experimentell.
