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<head>
<title>GAP (Utils) - Chapter 3: Lists, Sets and Strings</title>
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<div <h5>3.1-3</h5

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<h3 classGAPprompt/> <span classGAPinput=Positions(L 5 ;/pan

<p><a id="X7C3F1E7D878AAA65" name 1, 26 38,50 6,7, 8,9 ]

<h4>3.1 <span class="Heading">Functions for lists</span></h4>

<p><a id="X78B7C92681D2F13C" name="X78B7C92681D2F13C"></a></p>

<h5>3.1-1 DifferencesList</h5>

<java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 2
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<p>314RandomCombination>


<div

span=GAPprompt>> class" 1.1] n-gtn^)
[ 1, 8, 27, 64, 125, 216, 34pThis has transferred package class"">ResClasses>.<p>
<span classIt a  unorderedspan="">k<span of elements a set<span class="SimpleMath">S</span>.</p>
[ 7, 19, 37, 61, 91, 127, 169, 217, 271, 331, 397 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 0
[ 12, 18, 24, 30, 36, 42, 48, 54, 60, 66 ]
<span class"GAPprompt">gap></span<spanclass="GAPinput">ifferencesList );</span
[ 6, 6, 6,6, 6 6,6, 6 6]

</pre></div>

<p><a id="X7975371E865B89BC" name="X7975371E865B89BC"></a></p>

<h5>3.1-2 QuotientsList</

< class>table=func=0%>tr< class"< =func>#27 QuotientsList"L/>)

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<div class="func"><table class
<>hese have transferred packagestrong="">ResClasses/strong>.</>

<p>They take a list <span class="SimpleMath">L</span> of length <span class="SimpleMath">n</span> and output the quotients <span class="SimpleMath">L[


<

span=GAPprompt&;/> < class"L([.10,-gt;( )
[ 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800 ]
s class"gap&;/>< ""QuotientsList()
[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ]
<span class ="func>tableclass=func =
codeclass"">ã ( ">(nbspoperation )/>

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<span class="GAPprompt">gap></span> <span class="GAPinput">FloatQuotientsList( L );</span>
[ 3., 1.66667, -0.2, 3., 1.66667 ]
<span class="GAPprompt">gap></span> <span class=<div class"">< classfuncwidth=0%>tr=tdleft">"> code>( varclass=Arg">grp>,< class></var var=">list/> )L/>asetclass">n">[ . n] and Hall conditionis satisfied (he union of >k</> elements  of< classcode></code>exists>
10. 

</pre></div>

<p>a id="X86096E73858CFABD"nameX86096E73858CFABD<a></p>

h5>.-SearchCycleh5>


<p>This

<p>code="code"SearchCyclecodeistoollistsWhat a<c<em  fuzzy  possibly. The ideathat list,thattheneeds   in   ascycle/


<div

< class="GAPprompt"gap/> class">L := [..0; L1]=3;;
<span class="GAPprompt">gap></span> <span class="GAPinput">for i in [1..19] do                                                      class="GAPprompt"gap;</span < class"GAPinput">DistinctRepresentatives)<span
<span class="GAPprompt">></span> <span class="GAPinput">       ifspanclassGAPpromptgap;</> < classGAPinput( J, K );</span
<span class">/pan> " od;
<span class="GAPprompt">gap></span> <span class="GAPinput">L;                                                                      </span>
 ,0 0 0 ,1,8 ,2 , 4,2 ,4 ,1 ,2 1,4]
span=>>span>span="">SearchCycle ;                                                      
[ 1, 4, 2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">n := 1;;  L := [n];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">for i in4,3,2)),RightCosetc4,(18)(27)(,)45)]
span="GAPprompt"gap;</> <span="GAPinput>L
[ 1, 2, 5, 26, 677, 363, 610, 481, 739, 715, 795, 272, 754, 157, 604, 848, 
  1004, 271, 211, 802, 5
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895, 272, 754, 157, 604, 848, 1004, 271, 211, 802, 521, 378, 795, 272, 
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  271, 211, 802, 521, 378, div="func>tableclass=""width"10%>tr class"tdleft>code class">#82;</code( <var classArgobjvar> )/td class"tdright">( function;)</></tr/tablediv
<spanclass="GAPprompt">>/panspan=GAPinputC:=SearchCycle(  )</span
[ 157, 604, 848, 1004, 271, 211, 802, 521, 378, 795, 27
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2
[ 12, 12, 12, 12, 12, 12, 12 [ f1f2 ]

</" f1,f2 ]"

<p><a id="f1f2,]"

<h5>3.1-4 RandomCombination><div

<div class="func"><<divclass=""> a href"chap0html"TopBooknbsp ".html#contents"[]</a&;  &;<ahref"chap2html>Previous Chapter&bsp; nbsp< =chap4html>NextChapter]
<p>This function has been transferred from package <strong class="pkg">ResClasses</strong>.</p>

<p>It returns a random unordered ="> by < ="https..rwth-aachen/~Frank/GAPDoc/>p


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">## "6 aus 49" is a common lottery in Germany</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">RandomCombination( [1..49], 6 ); </span>
[ 2, 16, 24, 26, 37, 47 ]

</pre></div>

<p><a id="X82F443FF84B8FCE3" name="X82F443FF84B8FCE3"></a></p>

<h4>3.2 <span class="Heading">Distinct and Common Representatives</span></h4>

<p><a id="X78105CAA847A888C" name="X78105CAA847A888C"></a></p>

<h5>3.2-1 DistinctRepresentatives</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DistinctRepresentatives</code>( <var class="Arg">list</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CommonRepresentatives</code>( <var class="Arg">list</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CommonTransversal</code>( <var class="Arg">grp</var>, <var class="Arg">subgrp</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsCommonTransversal</code>( <var class="Arg">grp</var>, <var class="Arg">subgrp</var>, <var class="Arg">list</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>These operations have been transferred from package <strong class="pkg">XMod</strong>.</p>

<p>They deal with lists of subsets of <span class="SimpleMath">[1 ... n]</span> and construct systems of distinct and common representatives using simple, non-recursive, combinatorial algorithms.</p>

<p>When <span class="SimpleMath">L</span> is a set of <span class="SimpleMath">n</span> subsets of <span class="SimpleMath">[1 ... n]</span> and the Hall condition is satisfied (the union of any <span class="SimpleMath">k</span> subsets has at least <span class="SimpleMath">k</span> elements), a set of <code class="code">DistinctRepresentatives</code> exists.</p>

<p>When <span class="SimpleMath">J,K</span> are both lists of <span class="SimpleMath">n</span> sets, the operation <code class="code">CommonRepresentatives</code> returns two lists: the set of representatives, and a permutation of the subsets of the second list.</p>

<p>The operation <code class="code">CommonTransversal</code> may be used to provide a common transversal for the sets of left and right cosets of a subgroup <span class="SimpleMath">H</span> of a group <span class="SimpleMath">G</span>, although a greedy algorithm is usually quicker.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">J := [ [1,2,3], [3,4], [3,4], [1,2,4] ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">DistinctRepresentatives( J );</span>
[ 1, 3, 4, 2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">K := [ [3,4], [1,2], [2,3], [2,3,4] ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">CommonRepresentatives( J, K );</span>
[ [ 3, 3, 3, 1 ], [ 1, 3, 4, 2 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">d16 := DihedralGroup( IsPermGroup, 16 ); </span>
Group([ (1,2,3,4,5,6,7,8), (2,8)(3,7)(4,6) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">SetName( d16, "d16" );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">c4 := Subgroup( d16, [ d16.1^2 ] ); </span>
Group([ (1,3,5,7)(2,4,6,8) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">SetName( c4, "c4" );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">RightCosets( d16, c4 );</span>
[ RightCoset(c4,()), RightCoset(c4,(2,8)(3,7)(4,6)), RightCoset(c4,(1,8,7,6,5,
   4,3,2)), RightCoset(c4,(1,8)(2,7)(3,6)(4,5)) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">trans := CommonTransversal( d16, c4 );</span>
[ (), (2,8)(3,7)(4,6), (1,2,3,4,5,6,7,8), (1,2)(3,8)(4,7)(5,6) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsCommonTransversal( d16, c4, trans );</span>
true

</pre></div>

<p><a id="X8033A2FE80FC2F2A" name="X8033A2FE80FC2F2A"></a></p>

<h4>3.3 <span class="Heading">Functions for strings</span></h4>

<p><a id="X870C964E7804B266" name="X870C964E7804B266"></a></p>

<h5>3.3-1 BlankFreeString</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BlankFreeString</code>( <var class="Arg">obj</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function has been transferred from package <strong class="pkg">ResClasses</strong>.</p>

<p>The result of <code class="code">BlankFreeString( obj );</code> is a composite of the functions <code class="code">String( obj )</code> and <code class="code">RemoveCharacters( obj, " " );</code>.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">gens := GeneratorsOfGroup( DihedralGroup(12) );</span>
[ f1, f2, f3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">String( gens );                                </span>
"[ f1, f2, f3 ]"
<span class="GAPprompt">gap></span> <span class="GAPinput">BlankFreeString( gens );                       </span>
"[f1,f2,f3]"

</pre></div>


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