<p>Here we include a list of all functions that are provided to the User.</p>
<p>The following functions are designed to improve the User experience when working or experimenting with wreath products of finite groups and their elements. Most functions are about presenting elements in an intuitive way and giving access to useful information.</p>
<p>The main way for the User to look at a wreath product in a "human-readable" way is by using an isomorphism from a specialised wreath product representation to a generic representation.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsomorphismWreathProduct</code>( <var class="Arg">G</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns an isomorphism from a specialized wreath product <var class="Arg">G</var> to a generic wreath product.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">K := AlternatingGroup(5);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">H := SymmetricGroup(4);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := WreathProduct(K, H);</span>
<permutation group of size 311040000 with 10 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">iso := IsomorphismWreathProduct(G);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">W := Image(iso);</span>
<group of size 311040000 with 4 generators>
</pre></div>
<p>For an overview on wreath product representations in <strong class="pkg">GAP</strong> see <a href="chap5.html#X787C52A28302EB30"><span class="RefLink">5.1-1</span></a>.</p>
<p>In the background, it uses the low-level functions <code class="code">ListWreathProductElement</code> and <code class="code">WreathProductElementList</code> and wraps the <code class="code">IsList</code> representations into <code class="code">IsWreathProductElement</code> representations.</p>
<p>For performant code, we recommend to use these low-level functions instead of <code class="code">IsomorphismWreathProduct</code>. All functions for <code class="code">IsWreathProductElement</code> also work on <code class="code">IsList</code> objects that represent a wreath product element. However, it is not checked that the <code class="code">IsList</code> object actually represents a wreath product element.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ComponentsOfWreathProduct</code>( <var class="Arg">W</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns a list of two groups [K, H], where <var class="Arg">W = K wr H</var>. The argument <var class="Arg">W</var> must be a wreath product (see <a href="chap2.html#X7DF2AEBC8518FFA4"><span class="RefLink">2.1</span></a>).</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TopGroupOfWreathProduct</code>( <var class="Arg">W</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns a group, namely the top group <span class="SimpleMath">⟨ 1_K ⟩^m × H</span> of the wreath product <span class="SimpleMath">W = K ≀ H</span> (see <a href="chap2.html#X7DF2AEBC8518FFA4"><span class="RefLink">2.1</span></a>).</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BaseGroupOfWreathProduct</code>( <var class="Arg">W</var>[, <var class="Arg">i</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns a group, namely the base group <span class="SimpleMath">K^m × ⟨ 1_H</span> of the wreath product <span class="SimpleMath">W = K ≀ H</span>. If the optional argument <var class="Arg">i</var> is provided, the function returns the <var class="Arg">i</var>-th factor of the base group of <var class="Arg">W</var> (see <a href="chap2.html#X7DF2AEBC8518FFA4"><span class="RefLink">2.1</span></a>).</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TopComponentOfWreathProductElement</code>( <var class="Arg">x</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns a group element, namely the top component of <var class="Arg">x</var>. The argument <var class="Arg">x</var> must be a wreath product element (see <a href="chap2.html#X7DF2AEBC8518FFA4"><span class="RefLink">2.1</span></a>).</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BaseComponentOfWreathProductElement</code>( <var class="Arg">x</var>[, <var class="Arg">i</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns a group element, namely the base component of <var class="Arg">x</var>. If the optional argument <var class="Arg">i</var> is provided, the function returns the <var class="Arg">i</var>-thbase component of <var class="Arg">x</var>. The argument <var class="Arg">x</var> must be a wreath product element and the optional argument <var class="Arg">i</var> must be an integer (see <a href="chap2.html#X7DF2AEBC8518FFA4"><span class="RefLink">2.1</span></a>).</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Territory</code>( <var class="Arg">x</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a list, namely the territory of <var class="Arg">x</var>. The argument <var class="Arg">x</var> must be a wreath product element (see <a href="chap2.html#X83A8F3308644ACFA"><span class="RefLink">2.2</span></a>).</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsWreathCycle</code>( <var class="Arg">x</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns true or false. Tests whether <var class="Arg">x</var> is a wreath cycle. The argument <varclass="Arg">x</var> must be a wreath product element (see <a href="chap2.html#X83A8F3308644ACFA"><span class="RefLink">2.2</span></a>).</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSparseWreathCycle</code>( <var class="Arg">x</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns true or false. Tests whether <var class="Arg">x</var> is a sparse wreath cycle. The argument <var class="Arg">x</var> must be a wreath product element (see <a href="chap2.html#X872BE72F7B2BBD6B"><span class="RefLink">2.3</span></a>).</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ WreathCycleDecomposition</code>( <var class="Arg">x</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a list containing wreath cycles, namely the wreath cycle decomposition of <var class="Arg">x</var>. The argument <var class="Arg">x</var> must be a wreath product element (see <a href="chap2.html#X83A8F3308644ACFA"><span class="RefLink">2.2</span></a>).</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Yade</code>( <var class="Arg">x</var>[, <var class="Arg">i</var>] )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a group element, namely the yade of the wreath cycle <var class="Arg">x</var> evaluated at the smallest territory point. If the optional argument <var class="Arg">i</var> is provided, the function returns the yade evaluated at the point <var class="Arg">i</var>. The argument <var class="Arg">x</var> must be a wreath cycle and the optional argument <var class="Arg">i</var> must be an integer from the territory of <var class="Arg">x</var> (see <a href="chap2.html#X872BE72F7B2BBD6B"><span class="RefLink">2.3</span></a>)</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ViewObj</code>( <var class="Arg">x</var> )</td><td class="tdright">( method )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PrintObj</code>( <var class="Arg">x</var> )</td><td class="tdright">( method )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Display</code>( <var class="Arg">x</var>[, <var class="Arg">optrec</var>] )</td><td class="tdright">( method )</td></tr></table></div>
<p>Wreath product elements are viewed, printed and displayed (see Section <a href="../../../doc/ref/chap6.html#X8074A8387C9DB9A8"><span class="RefLink">Reference: View and Print</span></a> for the distinctions between these operations) as generic wreath product elements (see Section <a href="chap2.html#X7DF2AEBC8518FFA4"><span class="RefLink">2.1</span></a>). For an example of the distinctions and outputs see <a href="chap3.html#X7C85A3718343D842"><span class="RefLink">3.2</span></a>.</p>
<p>The method <code class="code">Display</code> allows an optional argument <var class="Arg">optrec</var> which must be a record and modifies the display output for the execution of a single instance of the command.</p>
<p>For modifying the display output globally for all subsequent executions of <code class="code">Display</code> see <code class="func">SetDisplayOptionsForWreathProductElements</code> (<a href="chap4.html#X7F900588870902F7"><span class="RefLink">4.4-3</span></a>).</p>
<p>The following components of <var class="Arg">optrec</var> are supported. Note, that in the following <em>labels</em> refer to the the printing output <q><span class="SimpleMath">1, /ldots, m</span></q> and <q>top</q> as seen in the tutorials.</p>
<dl>
<dt><strong class="Mark"><code class="code">horizontal</code></strong></dt>
<dd><p><code class="keyw">true</code> to use the horizontal printer. <em>DEFAULT</em></p>
<p><code class="keyw">false</code> to use the vertical printer.</p>
</dd>
<dt><strong class="Mark"><code class="code">labels</code></strong></dt>
<dd><p><code class="keyw">true</code> to print labels. <em>DEFAULT</em></p>
<p><code class="keyw">false</code> to suppress labels.</p>
</dd>
<dt><strong class="Mark"><code class="code">labelStyle</code></strong></dt>
<dd><p><code class="keyw">"none"</code> for labels in normal intensity. <em>DEFAULT</em></p>
<p><code class="keyw">"bold"</code> for labels in increased intensity.</p>
<p><code class="keyw">"faint"</code> for labels in decreased intensity.</p>
</dd>
<dt><strong class="Mark"><code class="code">labelColor</code></strong></dt>
<dd><p><code class="keyw">"default"</code> for labels in the default GAP output color. <em>DEFAULT</em></p>
<p><code class="keyw">"red"</code> for labels in red color.</p>
<p><code class="keyw">"blue"</code> for labels in blue color.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DisplayOptionsForWreathProductElements</code>( )</td><td class="tdright">( function )</td></tr></table></div>
<p>prints the current global display options for wreath product elements.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SetDisplayOptionsForWreathProductElements</code>( <var class="Arg">optrec</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>sets the current global display options for wreath product elements.</p>
<p>The argument <var class="Arg">optrec</var> must be a record with components that are valid display options. (see <a href="chap4.html#X8227AD6B7FC637DE"><span class="RefLink">4.4</span></a>) The components for the current global display options are set to the values specified by the components in <var class="Arg">optrec</var>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ResetDisplayOptionsForWreathProductElements</code>( )</td><td class="tdright">( function )</td></tr></table></div>
<p>resets the current global display options for wreath product elements to default.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CycleIndexWreathProductImprimitiveAction</code>( <var class="Arg">K</var>, <var class="Arg">H</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>For two permutation groups <var class="Arg">K</var> and <var class="Arg">H</var> this function constructs the cycle index polynomial of the wreath product <span class="SimpleMath">K ≀ H</span> in imprimitive action.</p>
<p>The implementation is based on <a href="chapBib.html#biBPolya">[P\t37]</a>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CycleIndexWreathProductProductAction</code>( <var class="Arg">K</var>, <var class="Arg">H</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>For two permutation groups <var class="Arg">K</var> and <var class="Arg">H</var> this function constructs the cycle index polynomial of the wreath product <span class="SimpleMath">K ≀ H</span> in product action.</p>
<p>The implementation is based on <a href="chapBib.html#biBHarrisonHigh">[HH68]</a> and <a href="chapBib.html#biBPalmerRobinson">[PR73]</a>.</p>
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