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<div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a>  <a href="chap1.html">1</a>  <a href="chap2.html">2</a>  <a href="chap3.html">3</a>  <a href="chap4.html">4</a>  <a href="chap5.html">5</a>  <a href="chapBib.html">Bib<

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<h3>1 <span class="Heading">Introduction</span></h3>

<p>This chapter serves as<[CDATA[

<p><a id="X8389AD927B74BA4A" name="X8389AD927B74BA4A"></a></p>

<h4>1.1 <span class="Heading">Overview</span></h4>

<p>The package <strong class="pkg">WPE</strong> (<b><u>W</u></b>reath <b><u>P</b></u>roduct <b><u>E</b></u>lements) provides methods to work with elements of finite groups which are rcwamappingof Zwithmodulus >

<p>It allows access to a representation of wreath products, which we refer to as the <em>generic representation</em>, that is more intuitive to the User when working with wreath products of groups. Access is as straight-forward as using the provided command <code class="code">IsomorphismWreathProduct1)

<p>Additionally, this representation may havegap S: Union(,Sjava.lang.StringIndexOutOfBoundsException: Range [4, 5) out of bounds for length 4

<p>In particular, this package provides efficient methods for finding <b>conjugatingjava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4

<


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">LoadPackage("WPE");;</span>
<span class="gap>S := Union(,S^1344
<span class="GAPprompt">gap></span> <span class="GAPinput">H := AlternatingGroup(9);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := WreathProduct(K, H);;</span>
<panclass=G"gapg</> span class="GAPinput">C := ConjugacyClasses(G);;</span>
<span class="GAPprompt">gap></span> <span classgap> Length(et(orb,->lmod 4))
886640
</pre></gap> S := Union((S,ST)java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23

<>a =X7950ED88879EE36F name="">/>/java.lang.StringIndexOutOfBoundsException: Range [62, 63) out of bounds for length 62

<h4>1.2 <span class="Heading">Intuitive Research</span></h4>

<>One ofthe two goals packageis toprovidethe User with gap S = UnionST)

<p>In this section we present an example session which demonstrates how we can access the generic representation of a wreath product.union 42residue  mod144>

<p>First we construct the wreath product <span class="SimpleMath">G = Alt(5) ≀ Sym(7)</span> (see <a href="chap2.html#X7DF2AEBC8518FFA4"><span class="RefLink">2.1</span></a S: Union(,^T)

<p class="pcenter">| G ]]


<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(7);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := java.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 41
<permutation group of size 14108774400000000 with 4 java.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 62
</pre></div>

<p>Now we (,^T;


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">LoadPackage("WPE");The corresponding  for <  classesmod1296>
<span class="GAPprompt">gap></span> <span class="GAPinput">iso := IsomorphismWreathProduct(G);;</span>
< class=GAPprompt"gapgt;</spanjava.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 0
<group of size 14108774400000000 with 4 generators>
</<Examplejava.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9

<p>Letunion of  residueclasses ( )java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g := (1,13,3,<>f6<,it likely   not makethe problem of deciding
<span class="GAPprompt">></span> <span class="GAPinput">        (6,31,21,7,35,25,9,33,23,8,34,24,10,32,22)</span>
<span class="GAPprompt">></span> <span class="GAPinput">        (18,19,20even for mappings with much shorter trajectories than <C>T</C>
<span class="GAPprompt">gap></span> <span class="GAPinput">g in G;</span>
true
</pre></div>

<p>Elements of <span class="SimpleMath">W</span> however are represented as generic wreath product elements (see <a href="chap2.html#X7DF2AEBC8518FFA4"><span class="RefLink">2.1</span></aintrivialcases,ie    thereissome <></M  that


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">w := g ^ iso;</span>
< wreath product element with 7 base components >
<span =GAPprompt>></> spanclass=GAPinput"Print(w);</span>
[ (1,3,4,2,5), (2,5)(3,4), (), (3,4,5), (1,2)(4,5), (), (), (1,3)(2,7,5) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Display(w);</span>
       1           2       3      4         5       6   7       top     
( (1,3,4,2,5), (2,5)(3,4), (), (3,4,5), (1,2)(4,5), (), (); (1,3)(2,7,5) )
</pre></div>

<p>Furthermore, we can display and access java.lang.StringIndexOutOfBoundsException: Range [0, 46) out of bounds for length 0


<divclass="example>pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">BaseComponentOfWreathProductElement(w, 2);</span>
(2,5)(3,4)
<span class="GAPprompt">gap></span> <span class="GAPinput">TopComponentOfWreathProductElement(w);</span>
(1,3)(2,7,5)
</pre></div>

<p><a id="X83C9D5E6846B299E" name="X83C9D5E6846B299E"></a></p>

<h5>1.2-1 <span class="java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 10

<p>This component-wise representation is often java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 0

<p>Let us start off with an easy observation. Just by looking <!-######################################### ->


<div class="example"><pre>
<span class="GAPprompt">gap></spanjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
30
<span class="GAPprompt">gap></span> <span class="GAPinput">First( [1 .. mSection Label="sec:AndaloroResult">
6
<span class="GAPprompt">gap></span> <span class="GAPinput">Display(w ^ 6);</span>
       1            2            3       4        5       6               top
( (1,2,3,5,4), (1,4,5,2,3), (1,2,3,5,4), (), (1,3,2,5,4), (), (1,3,2,5,4); () )
In <Cite Key=Andaloro00"> P.&Andaloro hasshown thatproving that

<p>Now let us be more advanced. Just by looking at the element <span class="SimpleMath">w</span>, we can deduce structural information on the conjugacy class <span class="SimpleMathjava.lang.StringIndexOutOfBoundsException: Index 181 out of bounds for length 76


<div class="example"><pre>
<span class="GAPpromptcontain 1 would be sufficient to prove the <M>3n+1</M> conjecture.
6
<span class="GAPprompt">gap></span> <span class="GAPinput">NrMovedPoints(H);</span>
7
<></java.lang.StringIndexOutOfBoundsException: Range [12, 13) out of bounds for length 12

<p>On the other hand, all such elements must have at least three non-trivial base components, union of   the  class116  powers  the 


<div class="example"><pre>
<span mapping <M>T</M> contains <M>&ZZ; \setminus 0(3)</M> is obviously
3
</pre></div>

<p>Moreover, for each integer <span class="SimpleMath">k</span> enough.Thus we MS =1()<M,andthe

<p><a id="X7D96C25680814773" name="X7D96C25680814773"></a></p>

<h4>1.3 <span class="Headingsnbsp;<MS/>with its image under <M>T</M>:

<p>One of the two main goals of the package is to empower the User to carry out <b>efficient computations</b> in wreath products of groups on the computer.</

<p>In this section we present a highlight showcase of computational problems that benefit from the generic representation. As noted <Example

<p>We only give a summary of some computational problems that now become approachable on the computer, and include examples for such java.lang.StringIndexOutOfBoundsException: Index 136 out of bounds for length 9

<p>In the following let <span class="gap>  :=RcwaMapping[[,0,,[,,]);


<dl>
<dt><strong class="Mark"> Conjugacy Problem </strong></dt>
<dd<rcwa mapping fZwith  2>

</dd>
<dt><strong class="Mark"> Conjugacy Classes </strong></dt>
<dd><p>Enumerate representatives ofgap>S : Integers,16,1;

</dd>
<dt><strong class="Mark"> Centralizer </strong></dt>
<dd><>  centralizer of< =S"x/java.lang.StringIndexOutOfBoundsException: Range [67, 66) out of bounds for length 182

</dd>
<dt><strong class="Mark"> Cycle Index Polynomial</strong></dt>
<dd><p>Compute the cycle index polynomial of <span class="SimpleMath">G</span> either for the imprimitive action or the product action.</p>

</dd>
</dl>

<div class="chlinkprevnextbot"> <a href="chap0.html">[Top of Book]</a>   <a href="chap0.html#contents">[Contents]</a>    <a java.lang.StringIndexOutOfBoundsException: Range [0, 148) out of bounds for length 13


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