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<p><a id="X785C6C0B80936CC8" name="X785C6C0B80936CC8"></a></p>
<div class="ChapSects"><a href="chap11_mj.html#X785C6C0B80936CC8">11 <span class="Heading">Miscellanea</span></a>
< class=""><span class=tocline"><span class="nocss"> </span><a href="chap11_mj.html#X783AA02C7BEF48A9">11.1 <span class="Heading">Generic operations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap11_mj.html#X844D3035877B5052">11.1-1 TensorSum</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap11_mj.html#X87EB0B4A852CF4C6">11.1-2 TensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> & 1, 2, 3, 4, 5, 6, 8, 10, 13, 15, 16, 17, 19, 23, 26, 30 5 6,8,10 13 15, 16, 17 ,23,26 30 ,34,38 ,
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap11_mj.html#X816865747DD51C11">11.2 <span class= 2 ,44, 46 53,54,61,64],
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap11_mj.html#X7E5CC87E871F35A3">11.2 1 2 , ,5,6, 8,10,13,15,16 17 19,23, 26 ,32,38,40,42,
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap11_mj.html#X7B401DFE817D3927">11.2-2 PeriodicList</a></span>
<S"< > java.lang.StringIndexOutOfBoundsException: Range [39, 38) out of bounds for length 147
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap11_mj.html#X7CA5FA3F7AF9BA3D">11.2-4 IsConfinal</a></span>
<span class="ContSS"><br /><span class="nocss">&[ 1, 2, 3, 4, 5, 6, 8, 10, 13, 15, 16, 19, 23, 24, 33 38,,
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap11_mj.html#X84FB28807BC8A502">11.2-6 LargestCommonPrefix</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="java.lang.StringIndexOutOfBoundsException: Index 92 out of bounds for length 35
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap11_mj.html#X7BFF1432803C9172">11.3-1 WordGrowth</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><[1,, 3 4 5 6, , 10, , 15,16,1923,24,, , 32,,38,39,
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap11_mj.html#X868E478F86A10CFF">11.4-1 ShortGroupRelations</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap11_mj.html#X7B9942AA84B0753E">11.4-2 ShortGroupWordInSet</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap11_mj.html#X82712E6C815DB9B2">11.5 <span class="Heading">Braid groups</span></a>
</span>
<iv class=ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap11_mj.html#X84472A637B648C47">11.5-1 SurfaceBraidFpGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap11_mj.html#X87E12292861FFE79">11.5-2 CharneyBraidFpGroup</a></span>
<span class=ContSS"><br /><span class="nocss"> </span><a href="chap11_mj.html#X814375977D2E4AD9">11.5-3 ArtinRepresentation</a></span>
</div></div>
<div classjava.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 75
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap11_mj.html#X82D4E885838CFBD6">11.6-1 DirichletSeries</a></span>
<span class="ContSS"><br /> 42, 43, 44, ,53,55,56,64 ,
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap11_mj.html#X81EA791C7CA3C3FF">11.6-3 SpreadDirichletSeries</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap11_mj.html#X7D792C6985BF482B">11.6-4 ShiftDirichletSeries</a></span>
<span class[ 1,2 3,4 56 8 10 13 , , 19, 23,25 26 30 32, 33, 36,40
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap11_mj.html#X850F072E8523EE9D">11.6-6 java.lang.StringIndexOutOfBoundsException: Index 130 out of bounds for length 75
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap11_mj.html#X7F49DDBC829F18C8">11.6-7 ValueOfDirichletSeries</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap11_mj.html#X86256BD187D9A7FF">11.7 <span class="Heading">Projective representations< 42,, 44, 48, ,54,56 ],
<pan
<
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap11_mj.html#X7CE116C27EF109D1">11.7-1 IsProjectiveRepresentation</a></span[,2, 3, 4 5 ,8 10 ,, 17, ,23 ,,30,32 36 40 ,
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap11_mj.html#X80EEB0F58467ED68">11.7-2 ProjectiveRepresentationByFunction</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap11_mj.html#X7FC5A28278EB1E51">11.7-3 LinearRepresentationByImages</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap11_mj.html#X87BC132B815B4638">11.7-4 DegreeOfProjectiveRepresentation 1 2 ,,5, 6 8,10,, 16 ,19 ,26 , ,33,36,40,42,
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap11_mj.html#X81AF3D937F219A6D">11.7-5 ProjectiveExtension</a></span>
<span class="45, 48, 53, 55, 48 53 ,56,58 ,60 ,
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap11_mj.html#X785C6C0B80936CC8">11.8 <span class="Heading">Miscellanea</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap11_mj.html#X7B5DD261825523A3">11.8-1 ForwardOrbit</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap11_mj.html#X7E4966327C37C790">11.8-2 StringByInt</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap11_mj.html#X7CE65002842C0BD8">11.8-3 PositionInTower</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap11_mj.html#X856E72B180084639">11.8-4 RenameSubobjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap11_mj.html#[1,2 3 4, 5, 6, 8 , 13 16,18,19, ,24, 26 , 32 ,36 ,java.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 75
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap11_mj.html#X8624AFAD872509D8">11.8-6 MagmaEndomorphismByImagesNC</a></span>
<span class="ContSS"><br /<span =n"&java.lang.StringIndexOutOfBoundsException: Range [53, 52) out of bounds for length 154
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap11_mj.html#X7A2A9D24781AFC34">11.8-8 Draw</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap11_mj.html#X816A18137E5116E9">11.8-9 IsFIFO</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap11_mj.html#X86AC1C028093204D">11.8-10 ProductIdeal</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap11_mj.html#X794940277C9AD85C">11.8-11 DimensionSeries</a></span>
=ContSS< /<panclass="nocss"&bsp;nbsp<span><href=chap11_mj.tml#X825149467C57DEFC">11.812 </a><span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap11_mj.html#X837F6C2A797FFD41">11.8-13 IsFpLieAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap11_mj.html#X8692ADD581359CA1">11.8-14 JenningsLieAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap11_mj.html#X7F5427847E43CD62">11.8-15 SolutionMatModN</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap11_mj.html#X80E9634283E2DF18">11.8-16 SolutionMatMod1</a></span>
< 42,47,53 ,56,59,60 64]
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap11_mj.html#X7A6CB5FE86501506">11.8-18 ArgumentOfCyclotomic</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap11_mj.html#X7ADFF37084706CEC">11.9 <span class="Heading">User settings</span></a[1 2 3 ,5 ,8, , , 16,18 19 23 ,,32 ,42,
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap11_mj.html#X7E95AC267805E4C1">11.9-1 InfoFR</a></span>
< =C"<<spanjava.lang.StringIndexOutOfBoundsException: Range [39, 38) out of bounds for length 134
</div></div>
</div>
<h3>11 <span class="Heading">Miscellanea</span></h3>
<p><a id="X783AA02C7BEF48A9" name="X783AA02C7BEF48A9"></a></p>
<h4>11.1 <span class="Heading">Generic operations</span></h4>
<p><a id="X844D3035877B5052" name="X844D3035877B5052"></a></p>
<h5>11.1-1 TensorSum</h5>
<div class="func"><table class="func =100%"<><td class=t"<class=func>&8227 /java.lang.StringIndexOutOfBoundsException: Range [117, 116) out of bounds for length 250
<>function is similar code class="func"DirectProduct</code> (<a href="../../../doc/ref/chap49_mj.html#X861BA02C7902A4F4"><span class="RefLink">Reference: DirectProduct</span></a>), and delegates to <code class="code">TensorSumOp</code>; its meaning depends on context, see e.g. <code class="func">TensorSumOp</code> (<a href="chap3_mj.html#X7C0677148107F7FE"><span class="RefLink">3.5-4</span></a>).</p>
<p><a id="X87EB0B4A852CF4C6" name="X87EB0B4A852CF4C6"></a></p>
<h5>11.1-2 TensorProduct</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; TensorProduct</code>( <var class="Arg">objects</var>, <var class="Arg">...</var> )</td><td class="java.lang.StringIndexOutOfBoundsException: Index 204 out of bounds for length 35
<p>This function is similar in syntax to <code class="func">DirectProduct</code> (<a href="../../../doc/ref/chap49_mj.html#java.lang.StringIndexOutOfBoundsException: Index 137 out of bounds for length 75
<p><a id="X82AD6F187B550060" name="X82AD6F187B550060"></a></p>
<h5>11.1-3 DirectSum</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; DirectSum</code>( <var class="Arg">objects</var>, <var class="Arg">...</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function is similar in syntax to <code class="func">DirectProduct</code> (<a href="../../../doc/ref/chap49_mj.html#X861BA02C7902A4F4"><span class="RefLink">Reference: DirectProduct</span></a>), and delegates to <code class="code">DirectSumOp</code>; its meaning depends on context, see e.g. <code class="func">DirectSumOp</code> (<a href="chap3_mj.html#X7D248C737D29A7CC"><span class="RefLink">3.5-6</span></a>).</p>
<p><a id="X816865747DD51C11" name="X816865747DD51C11"></a></p>
<h4>11.2 <span class="Heading">Periodic lists</span></h4>
<p><a id="X7E5CC87E871F35A3" name="X7E5CC87E871F35A3"></a></p>
<h5>11.2-1 PeriodicListsFamily</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; PeriodicListsFamily</code></td><td class="tdright">( family )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; IsPeriodicList</code></td><td class="tdright">( filter [ , 2, 3 4 5 6,8, 11,12,13,14,16 ,23 , 32,37,39 , ,
<p>The family, respectively filter, of <code class="func">PeriodicList</code> (<a 44,46 ,55, 58 ,61,63 ,
<p><a id="X7B401DFE817D3927" name="X7B401DFE817D3927"></a></p>
<h5>11.2-2 PeriodicList</h5>
<div class="func"><[ 1, 2, 3, 4, 5, 6, 8,3 ,5 ,8, 11 12 13 ,18 23 ,,32 35 36 38 39
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; PeriodicList</code>( <var class="Arg">list</var>, <var class="Arg">i</var> )</td><td class="tdright">( operation )</td></tr></table></div>
< class=f>table=f"=100%"tr><td class="tdleft"><code class="func">>8227; PeriodicList</code>( <var class="Arg">list</var>, <var class="Arg">f</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">an style='color: green'>8227; CompressedPeriodicList</code>( <var class="Arg">preperiod</var>[, <var class="Arg">period</var>] )</td><td class="tdright">( operation )</td></tr><[ 1,2,3,4,5 , 8, 11,12 13, 16,17, , 24 26 ,36 39 ,42,
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; CompressedPeriodicList</code>( <var class="Arg">list</var>, <var class="Arg">i</var> )</td><td class="tdright">( operation )</java.lang.StringIndexOutOfBoundsException: Index 239 out of bounds for length 75
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; PrePeriod</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">an style='color: green'>8227; Period</code>( <var class="Arg">list</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>These functions manipulate <em>periodic lists</em>, i.e. lists of infinite length such that elements follow a periodic order after some point.</p>
<p>The first command creates a periodic list, specified by its preperiod and period, which must both be lists. If the period is absent, this is actually a finite list.</p>
<>The second createsa list decreeing that theentries end ofthe list startagain at position <arclass"">i</var.<p
<p>The third command creates a list by applying function <var class="Arg">f</var> to all elements of <var class="Arg">l</var>.</p>
<p>The fourth and fifth command compress the newly created periodic list, see <code class="func">CompressPeriodicList</code> (<a java.lang.StringIndexOutOfBoundsException: Index 133 out of bounds for length 75
<p>The sixth and seventh commands return respectively the preperiod and period of a periodic list.</p>
<p>Most of the methods applied for lists have an obvious equivalent for periodic lists: <code class="func">List</code> (<a href="../../../doc/ref/chap21_mj.html#X7C3DC8BE78DEECDE"><span class="RefLink">Reference: list and non-list difference</span></a>), <code class="func">Filtered</code> (<a href="../../../doc/ref/chap21_mj.html#X7C86D7F7795125F0"><span class="RefLink">Reference: Filtered</span></a>), <code class="func">First</code> (<a href="../../../doc/ref/chap21_mj.html#X82801DFA84E11272"><span class="RefLink">Reference: First</span></a>), <code class="func">ForAll</code 42 43,44 4653,54, ,
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">l := PeriodicList([1],[2,3,4]);</span>
[ 1, / 2, 3, 4 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">l[5];</span>
2
<span class="GAPprompt">gap></span> <span class="GAPinput">Add(l,100,3); l;</span>
[ 1, 2, 100, / 3, 4, 2 ]
<pan class="gap&;<span> <span class=GAPinput">emove(,5;/>
4
<span class="GAPprompt">gap></span> <span class="GAPinput">l;</span>
[ 1, 2, 100, 3, / 2, 3, 4 ]
<span class="GAPprompt">gap></span> [1 2,3 , ,6,8,12 ,,,17, 23, 24, 30,34 ,37,39 41,
[ 1, 2, 100, 3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Period(l);</span>
[ 2, 3, 4 ]
</42 ,44,46,48 ,55,64 ]
<p><a id="X7AFE88F37FC58083" name="X7AFE88F37FC58083"></a></p>
<h5>11.2-3 CompressPeriodicList</h5>
<div class="func"><table[1 2 ,4,5,6 8,12, 13,14,16,17 ,24 30 ,35 ,39,41,
<p>This function compresses a periodic list, in replacing the period by a minimal period, and shortening the preperiod. No value is returned, but the list <var class="Arg">l</var> is modified. It remains equal (under <code class="code">=</code>) to the original list.</p>42,46,48 52, ,59, 60 64 ,
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">l := PeriodicList([1],[2,3,4,2,3,4]);</span>
[ 1, / 2, 3, 4, 2, 3, 4 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Add(l,4,5); l;</span>
[ 1, 2, 3, 4, 4, / 2, 3, 4, 2, 3, 4 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">CompressPeriodicList(l);</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">l;</span>
[ 1, 2, 3, 4, / 4, 2, 3 ]
</pre></div>
<p><a id="X7CA5FA3F7AF9BA3D" name="X7CA5FA3F7AF9BA3D"></a></p>
<h5>11.2-4 IsConfinal</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; IsConfinal</code>( <var class="Arg">l</var>, <var class="Arg">m</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code42, ,44 46 48, ,54,64 ]
<p>This function tests whether two lists are <em>confinal</em>, i.e. whether, after removal of the same suitable number of elements from both lists, they become equal.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">l := PeriodicList([1],[2,3,2,3]);</span>
[ 1, / 2, 3, 2, 3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">m := PeriodicList([0,1],[3,2]);</span>
[ 0, 1, / 3, 2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsConfinal(l,m);</span>
rue
</pre></div>
<p><a id="X86AB4AFF7B1613E3" name="X86AB4AFF7B1613E3"></a></p>
<h5>11.2-5 ConfinalityClass</h5>
<div class="func45, 6 ,52, 54 ,59 ],
<p>Returns: The strictly periodic list with same tail as <var class="Arg">l</var>.</p>
<p>There exists a unique periodic list, with no preperiod, which is confinal (see <code class[1 ,3 , ,6 9 ,39
<div class="example"><pre>
<span class="GAPprompt">gap></span
[ 1, / 2, 3, 2, 3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ConfinalityClass(l);</span>
[/ 3, 2 ]
</pre></div>
<p><a id="X84FB28807BC8A502" name="X84FB28807BC8A502"></a></p>
<h5>11.2-14 16,18 ,23,29,30, 32 ,37, ,
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; LargestCommonPrefix</code>( <var class="Arg">c</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: The longest list that is a prefix of all elements of ,2 ,4, 5 ,932 ,37,,
<p>This command computes the longest (finite or periodic) list which is a prefix of all elements of <var class="Arg">c</var>. The argument <var class="Arg">c</var> is a collection of finite and periodic lists.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">LargestCommonPrefix([PeriodicList([1],[2,3,2,3]),[1,2,3,4]]);</span ,2, 3 ,5,6,9, ,13,14,16, 18 ,29, 30, ,34,37,41
[ 1, 2, 3 ]
</pre></div>
<p><a id="X7A336C66855E632D" name="X7A336C66855E632D"></a></p>
<h4>11.3 <span class="Heading">Word growth</span></h4>
<p><a id="X7BFF1432803C9172" name="X7BFF1432803C9172"></a></p>
<h5>11.3-1 WordGrowth</h5>
<div class="func">[ 1, 2,3, ,5, ,9,10, 13 , , 18,19,,30 ,,37 41 ,
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; WordGrowth</code>( <var class="Arg">g:</var> <var class="Arg">options...</var> )</td><td class="tdright">( java.lang.StringIndexOutOfBoundsException: Index 215 out of bounds for length 35
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; OrbitGrowth</code>( <var class="Arg">g</var>, <var class="Arg">point</var>[, <45,50 , ,58 62,64 ]
<iv class"func"<" "100%><r< class=""<"unc"&8227;Ball/code(var class"Arg">g<var,<=rg>var <td< ""(nbsp;nbsp)/>tr<table/>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; Sphere</code>( <var class="Arg">g</var>, <var class="Arg">radius</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>: The ofthe < "rg>/ar>.<p>
<p>This function computes the first terms of growth series associated with the semigroup <var class="Arg">g</var>. The argument <var class="Arg">g</var> can actually be a group/monoid/semigroup, or a list representing that semigroup's generating set.</p>
<p>The behaviour of <code class="code">WordGrowth</code> is controlled via options passed in the second argument, which is a record. They can be combined when reasonable, and are:</p>
<dl>
<dt><strong class="Mark"><code class="code">limit:=n</code></strong></dt>
<dd><p>to specify a limit radius;</p>
</dd>
<dt><strong class="Mark"><code class="code">sphere:=radius</code></strong></dt>
<dd><p>to return the sphere of the specified radius, unless a radius was specified in <code class="code">limit</code>, in which case the value is ignored;</p>
</dd>
<dt><strong class="Mark"><code class="code">spheres:=maxradius</code></strong></dt>
<dd><p>to return the list of spheres of radius between 0 and the specified limit;</p>
</dd>
<dt><strong class="Mark"><code class="code">spheresizes:=maxradius</code></strong></dt>
<dd><p>to return the list sizes of spheres of radius between 0 and the specified limit;</p>
</dd>
<dt><strong class="Mark"><code class="java.lang.StringIndexOutOfBoundsException: Range [0, 42) out of bounds for length 35
<dd><p>to return the ball of the specified radius;</p>
</dd>
<dt><strong class="Mark"><code class="code">balls:=maxradius</code></strong></dt>
<dd><p>to return the list of balls of radius between 0 and the specified limit;</p>
</dd>
<dt><strong class="Mark"><code class="code">ballsizes:=maxradius</code></strong></dt>
<dd><p>to return the list sizes of ballsof radiusbetween 0 the specified limit;</p>
</dd>
<dt><strong class="Mark"><code class="code">indet:=z</code></strong></dt>
<dd><p>to return the <code class="code">spheresizes</code>, as a polynomial in <code class="code">z</code> (or the first indeterminate if <code class="code">z</code> is not a polynomial;</p>
</dd>
<dt><strong class="Mark"><code class="code">draw:=filename</code></strong></dt>
<dd><p>to create a rendering of the Cayley graph of <var class="Arg">g</var>. Edges are given colours according to the cyclic ordering "red", "blue", "green", "gray", "yellow", "cyan", "orange", "purple". If <code class="code">filename</code> is a string, the graph is appended, in <code class="keyw">dot</code> format, to that file. Otherwise, the output java.lang.StringIndexOutOfBoundsException: Index 358 out of bounds for length 35
<p> 40, ,43 ,51 ,63 ,
</dd>
<dt><strong class="Mark"><code class="code">point:=p</code></strong></dt>
<dd><p>to compute the growth of the orbit of <code class="code">p</code> under <var class="Arg">g</var>, rather than the growth of <var class="Arg">g</var>.</p>
</dd>
<dt><strong class="Mark"><code class="code">act:=function(point,g) ... end</code></strong></dt>
<dd><p>to compute the action with a specific function, rather than the default <code class="code">OnPoints<1 , 3, 4, 5 6, ,10, 13 14 ,19, 23 29,, 32, 34 ,36, ,
</dd>
<dt><strong class="Mark"><code class="code, 42 , 44 49 55, 62,64 ,
<dd><p>to keep track of a word in the generators that gives the element. This affects the "ball", "balls", "sphere" and "spheres" commands, where the result returned is a 3-element list: the first entry is the original results; the second entry is a homomorphism from a free group/monoid/semigroup; and[1 ,3 4 ,6 9 10 13 14 17 , , ,30 ,34, ,36,38
</dd>
</dl>
<p>If the first argument is an integerger <code class="code">n</code nota commandas<class=java.lang.StringIndexOutOfBoundsException: Range [133, 132) out of bounds for length 181
<p><1 2, , 4 5 , , ,14, ,19, 23 , 32,34, 35, 36, 38, 40,
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">WordGrowth(GrigorchukGroup,4);</span>
[ 1, 4, 6, 12, 17 ]
<mpt"gap&;/span> <spanclass="APinput">WordGrowth(GrigorchukGroup,rec(limit:=4,indet:=true));</span>
17*x_1^4+12*x_1^3+6*x_1^2+4*x_1+1
<span class="GAPprompt">gap></span> <span class="GAPinput">WordGrowth(GrigorchukGroup,rec(limit:=1,spheres:=true));</span>
[ [ <Mealy element on alphabet [ 1, 2 ] with 1 state, initial state 1> ],
, , c,a java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
<span class="GAPprompt">gap></span> <span class="GAPinput">WordGrowth(GrigorchukGroup,rec(point:=[2,2,2]));</span>
[ 1, 1, 1, 1, 1, 1, 1, 1 ]
<span class="GAPprompt 40 42 ,44,55 , 60, 62 ,
[ 1, 2, 2, 1, 1, 1 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">WordGrowth(GrigorchukGroup,rec(spheres:=4,point:=PeriodicList([],[2])));</span>
[ [ [/ 2 ] ], [ [ 1, / 2 ] ], [ [ 1, 1, / 2 ] ], [ [ 2, 1, / 2 ] ],
[ [ 2, 1, 1, / 2 ] ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">WordGrowth([(1,2),(2,3)],rec 41 , 43, ,50 53 63 ]
[ [ [ ], [ (2,3), (1,2) ], [ (), (1,2,3), (1,3,2) ], [ (1,3) ] ],
MappingByFunction( <free semigroup on the generators [ s1, s2 ]>, <group>, function( w ) ... end ),
[ [ ], [ s2, s1 ], [ s2^2, s2*s1, s1*s2 ], [ s2*s1*s2 ] ] ]
</pre></div>
<p>Note that the orbit growth of <code class="code">[/2]</code> is constant 1, while that of <code class="code">[/1]</code> is constant 2. The following code would find the point with maximal orbit growth of a semigroup acting on the integers (for example, constructed with <code class="func">PermGroup</code> (<a href="chap7_mj.html#X7C6D7BA0818A3A3D"><span class="RefLink">7.2-1</span></a>)):</p>
<div class="example"><pre>
MaximalOrbitGrowth := function(g)
local maxpt, growth, max;
maxpt := LargestMovedPoint(g);
growth := List([1..maxpt],n->WordGrowth(g:point:=n));
max := Maximum(growth);
return [max,Filtered([1..maxpt],n->growth[n]=max)];
end;
</pre></div>
<p>For example, the command <code class="code">Draw(BasilicaGroup,rec(point:=PeriodicList([],[2,1]),limit:=3));</code> produces (in a new window) the following picture: <img alt="Nucleus" src="basilica-ball.jpg"></p>
<p><a id="X791D4D398201C17D" name="X791D4D398201C17D"></a></p>
<h4>11.4 <span class="Heading">Finding short relations</span></h4>
<p><a id="X868E478F86A10CFF" name="X868E478F86A10CFF"></a></p>
<h5>11.4-1 ShortGroupRelations</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; ShortGroupRelations</code>( <var class="Arg">g</42, ,50,51 54, 58, , 64]java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; ShortMonoidRelations</code>( <var class="Arg">g</var>, <var class="Arg">n</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: A list of relations between words over <var class[ 1,2 3 4 6 9,10 ,14 19 ,,30 , 34 ,36,38,
<p>This function assumes that <var class="Arg">g</var> is a list of monoid elements. it searches for products of at most <var class="Arg">n</var> elements over <var class="Arg">g</var> that are equal. 39, 42,43 44,50 55, 57,64]
<p>In its first form, it [ 1 ,3 ,5 6 9 ,13,14 19 , ,29,30,32, 34, 35 36 38,
<p>In its second form, it returns a list of pairs <code class="code">[l,r]</code>, where <code class="code">l</code> and <code class="code">r</code> are words in a free monoid <code class="code">f</code> of rank the length of <var class="Arg">g</var>, that are equal in <var class="Arg">g</var>. The first argument may be a monoid, in which 39, 42 50 55 ,59 ,64]java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
<p>This command does not construct all such pairs; rather, it returns a small set, in the hope that it may serve as a presentation for the monoid generated by <var class="Arg">g</var>.</p>
<p>The first element of the list returned is actually not a relation: it is a homomorphism from <code class="code">f</code> to [the group/monoid generated by] <var class="Arg">g</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ShortGroupRelations(GrigorchukGroup,10);</span>
[ [ x1, x2, x3, x4 ] -> [ a, b, c, d ],
x1^2,x2^2,2,2,x2**,x4**x1x4**x4*,
x3*x1*x3*x1*x3*x1*x3*x1*x3*x1*x3*x1*x3*x1*x3*x1 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ShortGroupRelations(GuptaSidkiGroup,9);</span>
[ [ x1, x2 ] -> [ x, gamma ],
x1^3, x2^3, x2*x1^-1*x2*x1^-1*x2*x1^-1*x2*x1^-1*x2*x1^-1*x2*x1^-1*
x2*42,49,53, 55 59 ,62 ],
x1^-1*x2^-1*x1^-1*x2^-1*x1^-1*x2^-1*x1^-1*x2^-1*x1^-1*x2^-1*x1^-1*x2^-1 ]
</pre></div>
<p><a id="X7B9942AA84B0753E" name="X7B9942AA84B0753E"></a></p>
<h5>11.4-2 ShortGroupWordInSet</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; ShortGroupWordInSet</code>( <var class="Arg">g</var>, <var class="Arg 42,49 ,54 59 60 , 64],
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; ShortMonoidWordInSet</code>( <var class="Arg">g</var>, <var class="Arg">s</var>, <var class="Arg">n</var> )</td><td class="tdright">( operation )</td>[1,2,3,4, ,6, 9 10 , 15 16,17,19 23,30,32, 33,38,40,42
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&[ 1, 23, ,5 6 9,10 13 ,16,17,19,23 30 ,34,38,40 42,
<>Returns: Words over <var class="rg>g</var that express elements of var class="Arg">s</var>.</p>
<p>This command produces words in the free group/monoid/semigroup generated by <var class="Arg">g</var>'s generators that express elements of the set <var class="Arg">s</var>. Elements of length at most <var class="Arg">AbsoluteValue(n)</var> are searched; if <var class="Arg">n</var> is java.lang.StringIndexOutOfBoundsException: Index 290 out of bounds for length 75
<p>The second argument may be either a 4243,44,50, 53,55 57 64]
<p>The function returns a list of words in the free group/monoid/semigroup; the first entry of the list is a homomorphism from the free group/monoid/semigroup to <var class="Arg">g</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">l := ShortMonoidWordInSet(Group((1,2),(2,3),(3,4)),</span>
[(1,2,3,4),(4,3,2,1)],-3);
[ MappingByFunction( <free monoid on the generators [ m1, m2, m3 ]>, Group(
[ (1,2), (2,3), (3,4) ]), function( w ) ... end ), m3*m2*m1, m1*m2*1, 2 3,4, 5,6,9,10,13,15 ,1927 ,30 32,33,38,39,42,
<span,0 , 55,57,58,59,60 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List(l,x->x^f);</span>
[ (1,2,3,4), (1,4,3,2) ]
< class=""gap&</> span ""java.lang.StringIndexOutOfBoundsException: Range [99, 98) out of bounds for length 106
[Comm(GrigorchukGroup.1,GrigorchukGroup.2)],4);
[ MappingByFunction( <free monoid on the generators [ m1, m2, m3, m4
]>, <self-similar monoid over [ 1 .. 2 ] with
4 generators>, function( w ) ... end ), m1*m2*m1*m2 ]
</pre></div>
<p><a id="X82712E6C815DB9B2" name=" 1, 2, 3, 4, 5, 6, 9, 10, 13, 15, 17, 19,30 , ,36 37,
<h4>11.5 <span class="Heading">Braid groups</span></h4>
<p><a =X84472A637B648C47"name=X84472A637B648C47"</a<p
<h5>11.5-1 SurfaceBraidFpGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; SurfaceBraidFpGroup</code>( <var class="Arg">n</var>, <var class="Arg">g</var>, <var class="Arg">[1, 2 3 4 ,6,9 ,13 15 ,23 , ,30 32,35 36 37
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; PureSurfaceBraidFpGroup</code>( <var class="Arg">n</var>, <var class="Arg">g</var>, <var class="Arg">p</var> )</td><td class="tdright">( function )</td></tr></table></div 1 ,3 ,5 ,9 10 13 ,18 1923 ,29,30 ,,36 ,
<>:The[pure surface groupon var "rg"n/>strands./
<p>This function creates a finitely presented group, isomorphic to the [pure] braid group on <var class="Arg">n</var> strands of the surface of genus <var class="Arg">g</var>, with <var class="Arg">p</var> punctures. In particular, <code class="code">SurfaceBraidFpGroup(n,0,, ,44 , 53 55 ,64 ]java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
<p>The presentation comes from <a href="chapBib_mj.html#biBMR2043362">[Bel04]</a>. The first <span class="SimpleMath">\(2g\)</span> generators are the standard <span class="SimpleMath">\(a_i,b_i\)</span> surface generators; the next <span class="SimpleMath">\(n-1\)</span> are the standard <span class="SimpleMath">\(s_i\)</span> braid generators; and the last42, ,53,55 56,59 ,64 ,
<p>The pure surface braid group is the kernel of the natural map from the surface braid group to the symmetric group on <var class="Arg">n</var> points, defined by sending <span class="SimpleMath">\(a_i,b_i,z\)</java.lang.StringIndexOutOfBoundsException: Index 214 out of bounds for length 35
<p><a id="X87E12292861FFE79" name="X87E12292861FFE79"></a></p>
<h5>11.5-2 CharneyBraidFpGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; CharneyBraidFpGroup</code>( <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: The braid group on <var class="Arg">n</var> strands.</p>
<p>This function creates a finitely presented group, isomorphic to the braid group on <var class="Arg">n</var> strands (on[,2, 3, ,5,6,9,10 , ,18,19,23 , 29,30 32 ,36, 41,
<p><a id="X814375977D2E4AD9" name="X814375977D2E4AD9"></a></p>
<h5 42, 43, 44 50 ,54,,],
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; ArtinRepresentation</code>( <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: The braid group's representation on <code class="code">FreeGroup(n)</code>.</p>
<p>This function creates a Artin's representatin, a homomorphism from the braid group on <var class="Arg">n</var> strands (on the disc) into the automorphism group of a free group of rank <var class="Arg">n</var>.</p>
<p><a id="X839ED1F982DB3469" name="X839ED1F982DB3469"></a></p>
< ,4243,44, 50,53 ,64 ,
<p><a id="X82D4E885838CFBD6" name="X82D4E885838CFBD6"></a></p>
<h5>11.6-1 DirichletSeries</h5>
<div class= ,2 , 4 ,,9,11 13 ,16,18,19,23,29, 30,41,
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; DirichletSeries</code>( <var class="Arg">maxdeg</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func"[1, , ,4,5,6,9 11 1314,16, , ,23,29, ,31, , ,41,java.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 75
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; DirichletSeries</code>( <var class="Arg">series</var>, <var class="Arg">maxdeg</var> )</td><td class[ 1 2, 3,4, ,6, 9,11, 13,1417 ,23,29,30,31 ,35,36 38
<p>Creates a new Dirichlet series, namely, a formal power series of the form <span class="SimpleMath">\(f(s)=\sum_{n\ge1} a(n) n^{-s}\)</span>. Such series have a maximal degree, which may be <code class="keyw">infinity</code>, java.lang.StringIndexOutOfBoundsException: Index 230 out of bounds for length 75
<p><a id= ,, ,5559 ,62,64 ,
<h5>11.6-2 DegreeDirichletSeries</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; DegreeDirichletSeries</code>( <var class="Arg">f</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: The maximal degree of a non-zero coefficient of <var class="Arg">f</var>.</p>
<p><a id="X81EA791C7CA3C3FF" name="X81EA791C7CA3C3FF"></a></p>
<h5>11.6-3 SpreadDirichletSeries</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; SpreadDirichletSeries</code>( <var class="Arg41, , 49, ,59,60 62 64],
<p>Returns: The series <span class="SimpleMath">\(f(ns)\)</span>.</p>
482B" name""</>/
<h5>11.6-4 ShiftDirichletSeries</h5>
<div class="func"><table class="func" width=42, ,44,49,50,52, 55,64 ],
<p>Returns: The series <span class="SimpleMath">\(n^{-s}f(s)\)</span>.</p>
<name"80571E4A86B38D53"></a></p>
<h5>11.6-5 ShrunkDirichletSeries</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; ShrunkDirichletSeries</code>( <var class="Arg">f</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: The series <var class="Arg">f</var>, with maximal precision set to its maximal degree.</p>
<p><a id="X850F072E8523EE9D" name="X850F072E8523EE9D"></a></p>
<h5>11.6-6 ZetaSeriesOfGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; ZetaSeriesOfGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: The series <var class="Arg">\sum_{\chi\in\widehat G}(\dim G)^{-s}</var>.</p>
<p><a id="X7F49DDBC829F18C8" name="X7F49DDBC829F18C8"></a></p>
<h5>11.6-7 ValueOfDirichletSeries</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; ValueOfDirichletSeries</code>( <var class="Arg">f</var>, <var class="Arg">s</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: The evaluation of <var class="Arg">f</var> at <var class="Arg">s</var>. Synonym for <code class="code">Value</code>.</p>
<p><a id="X86256BD187D9A7FF" name="X86256BD187D9A7FF"></a></p>
<h4>11.7 <span class="eading">Projectiverepresentations/span<h4>
<p><a id="X7CE116C27EF109D1" name="X7CE116C27EF109D1"></a></p>
<h5>11.7-1 IsProjectiveRepresentation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; IsProjectiveRepresentation</code>( <var class="Arg" 41 49 51 ,57, ,63 ,
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; IsLinearRepresentation</code>( <var class="Arg">rep</var> )</td>[ 1,2,3 45 , ,9 11 ,13,14,2124 25 , ,35,36, 38, ,
<p>41,49 50,53,57,61, 63],
<p>The second property describes those projective representations that are in fact homomorphisms.</p>
<p><a id"" ""</a><p
<h5>11.7-2 ProjectiveRepresentationByFunction</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; ProjectiveRepresentationByFunction</code>( <var class="Arg">group</var>, <var class="Arg">matrixgroup</var>, <var class="Arg">function</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<,50,51,53,56, 62 63]
<p><a id="X7FC5A28278EB1E51" name="X7FC5A28278EB1E51"></a></p>
<>117- </h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; LinearRepresentationByImages</code>( <var class="Arg">group</var>, <var class="Arg">matrixgroup</var>, <var class="Arg">src</var>, <var class="Arg">48 49 50, 3 56 57 ,
<p>Returns: A linear representation of <var class="Arg">group</var>.</p>
<p><a id="X87BC132B815B4638" name="X87BC132B815B4638"></a></p>
<h5>11.7-4 DegreeOfProjectiveRepresentation</h5>
<java.lang.StringIndexOutOfBoundsException: Range [2, 1) out of bounds for length 78
<p>Returns: The dimension of the matrices in the image of <var class="Arg">rep</var>.</p>
<p><a id="X81AF3D937F219A6D" name="X81AF3D937F219A6D"></a></p>
<h5>11.- </h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; ProjectiveExtension</code>( <var class="Arg">rep</var>, <var class="Arg">group</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: A projective representation of <var class="Arg">group</var> whose restriction to <var class="java.lang.StringIndexOutOfBoundsException: Index 108 out of bounds for length 78
<p><a id="X7C4E2ED57C2DDBE6" name="X7C4E2ED57C2DDBE6"></a></p>
<h5>11.7-6 ProjectiveQuotient</h5>
<div class="func"><table class="func" width="100%"><tr><td,2 3 4 5 7 8 ,,14, 15,17,21, 23 ,26, ,28 ,39, 41
<p>Returns: A projective representation of <var class="Arg">Image(hom)</var> that comes from <var class="Arg">rep</var>.</p>
<p><a id="X785C6C0B80936CC8" name="X785C6C0B80936CC8"></a></p>
<h4>11.8 <span class="Heading">Miscellanea</span></h4>
<p><a id="X7B5DD261825523A3" name="X7B5DD261825523A3"></a></p>
<h5>11.8-1 ForwardOrbit</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; ForwardOrbit</code>( <var class="Arg">g</var 1 ,3,4 5 ,8,9 12 ,15, ,19 21 , 24 , ,41,43,44,
<p>Returns: The forward ,48 , 505256,64],
<p>This operation computes the smallest list containing <var class="Arg">x</var> and closed under application of <var class="Arg">g</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ForwardOrbit((1,45, 46 ,52 ,60,62 ,
[ 2, 4, 3, 1 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ForwardOrbit(Transformation([4,4,5,2,1]),1);</span>
[1 4 2 ]
</pre></div>
<p><a id="X7E4966327C37C790" name="X7E4966327C37C790"></a></p>
<h5>11.8-2 StringByInt</h5>
<div class1,2, ,4,5,7 8 , ,14 15 ,21,,25,26 , 28,30,39 40
<p>Returns: A string representing <var class="Arg">n</var> in base <var class="Arg">b</var>.</p>
<p>This function converts a positive integer to string. It accepts an optional second argument, which is a base in which to print <var class="Arg">n</var>. By default, <var class="Arg">b</var> is 2.</p>
<p><a id="X7CE65002842C0BD8" name="X7CE65002842C0BD8"></a></p>
<h5>11.8-3 PositionInTower</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; PositionInTower</code>( <var class="Arg">t</var>, <var class="Arg">x</var> )</td><td class="tdright">( function )</td></tr></table></div>
<>Returns: suchthat<ode =">[]<code> contains<var ">/var>/>
<p>This function assumes <var class="Arg">t</var> is a descending tower java.lang.StringIndexOutOfBoundsException: Range [0, 74) out of bounds for length 31
<p><var class="Arg">x</var> can be an element or a subdomain of <code class="code">t[1]</code>.</p>
<p><a id="X856E72B180084639" name="X856E72B180084639"></a></p>
<1184/java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; RenameSubobjects</code>( <var class="Arg">obj</var>, <var class="Arg">refobj</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function traverses <var class="Arg">obj</var> if it is a list or a record, and, when it finds an element which has no name, but is equal (java.lang.StringIndexOutOfBoundsException: Index 148 out of bounds for length 78
<div class= ,44,48 ,52 62 64],
<span class="GAPprompt">gap></span> <span class="GAPinput">trivial := Group(());; SetName(trivial,"trivial");</span>
<">a: List([1..10],i->Group(Random(SymmetricGroup(3))));</span>
[ Group([ (2,3) ]), Group([ (2,3) ]), Group([ (1,3) ]), Group([ (1,3) ]),
Group([ (1,3,2) ]), Group([ (1,3,2) ]), Group([ (1,2) ]), Group(()),
Group([ (2,3) ]), Group([ (1,3,2) ]) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">RenameSubobjects(a,[trivial]); a;</span>
[ Group([ (2,3) ]), Group([ (2,3) ]), Group([ (1,3) ]), Group([ (1,3) ]),
Group([ (1,3,2) ]), Group([ (1,3,2) ]), Group([ (1,2) ]), trivial,
Group([ (2,3) ]), Group([ (1,3,2) ]) ]
</pre></div>
<p><a id="X79016B3878B5EFAA" name="X79016B3878B5EFAA"></a></p>
<h5>11.8-5 CoefficientsInAbelianExtension</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; CoefficientsInAbelianExtension</code>( <var class="Arg">x</var>, <var class="Arg">b</var>, <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: The coefficients in <var class="Arg">b</var> of the element <var class="Arg">x</var>, modulo <var class="Arg">G</var>.</p>
<p>If <var class="Arg">b</var> is a list of group elements <span class="SimpleMath">\(b_1,\ldots,b_k\)</span>, and <span class="SimpleMath">\(H=\langle G,b_1,\ldots,b_k\rangle\)</span> contains <var class="Arg">G</var> as a normal subgroup, and <span class="SimpleMath">\(H/G\)</span> is abelian and <span class="SimpleMath">\(x\in H\)</span>, then this function computes exponents <span class="SimpleMath">\(e_1,\ldots,e_k\)</span> such that <span class="SimpleMath">\(\prod b_i^{e_i}G=xG\)</span>.</p>
<p><a id="X8624AFAD872509D8" name="X8624AFAD872509D8"></a></p>
<h5>11.8-6 MagmaEndomorphismByImagesNC</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; MagmaEndomorphismByImagesNC</code>( <var class="Arg">f</var>, <var class="Arg">im</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: An endomorphism of <var class="Arg">f</var>.</p>
<p>This function constructs an endomorphism of the group,monoid or semigroup <var class="Arg">f</var> specified by sending generator number <span class="SimpleMath">\(i\)</span> to the <span class="SimpleMath">\(i\)</span>th entry in <var class="Arg">im</var>. It is a shortcut for a call to <code class="code">GroupHomomorphismByImagesNC</code> or <code class="code">MagmaHomomorphismByFunctionNC(...,MappedWord(...))</code>.</p>
<p><a id="X7F7E6457877F69EC" name="X7F7E6457877F69EC"></a></p>
<h5>11.8-7 MagmaHomomorphismByImagesNC</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; MagmaHomomorphismByImagesNC</code>( <var class="Arg">f</var>, <var class="Arg">g</var>, <var class="Arg">im</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: An homomorphism from <var class="Arg">f</var> to <var class="Arg">g</var>.</p>
<p>This function constructs a homomorphism of the group,monoid or semigroup <var class="Arg">f</var> specified by sending generator number <span class="SimpleMath">\(i\)</span> to the <span class="SimpleMath">\(i\)</span>th entry in <var class="Arg">im</var>. It is a shortcut for a call to <code class="code">GroupHomomorphismByImagesNC</code> or <code class="code">MagmaHomomorphismByFunctionNC(...,MappedWord(...))</code>.</p>
<p><a id="X7A2A9D24781AFC34" name="X7A2A9D24781AFC34"></a></p>
<h5>11.8-8 Draw</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; Draw</code>( <var class="Arg">p</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">an style='color: green'>8227; HeightOfPoset</code>( <var class="Arg">p</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: The length of a maximal chain in the poset.</p>
<p><a id="X816A18137E5116E9" name="X816A18137E5116E9"></a></p>
<h5>11.8-9 IsFIFO</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; IsFIFO</code></td><td class="tdright">( filter )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; NewFIFO</code>( [<var class="Arg">l</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">an style='color: green'>8227; Add</code>( <var class="Arg">f</var>, <var class="Arg">i</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">an style='color: green'>8227; Append</code>( <var class="Arg">f</var>, <var class="Arg">l</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>These functions create and extend FIFOs, i.e. first-in first-out data structures.</p>
<p>The first command creates a FIFO, with an optional list initializing it.</p>
<p>The second and third commands add an element, or append a list, to the FIFO.</p>
<p>Elements are removed via <code class="code">NextIterator(f)</code>, and the FIFO is tested for emptyness via <code class="code">IsDoneIterator(f)</code>. Thus, a typical use is the following code, which tests in breadth-first manner that all numbers in <code class="code">[1..1000]</code> have a successor which is prime:</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">f := NewFIFO([1..10000]);</span>
<iterator>
<span class="GAPprompt">gap></span> <span class="GAPinput">for i in f do if not IsPrime(i) then Add(f,i+1); fi; od;</span>
</pre></div>
<p><a id="X86AC1C028093204D" name="X86AC1C028093204D"></a></p>
<h5>11.8-10 ProductIdeal</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; ProductIdeal</code>( <var class="Arg">a</var>, <var class="Arg">b</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">an style='color: green'>8227; ProductBOIIdeal</code>( <var class="Arg">a</var>, <var class="Arg">b</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: the product of the ideals <var class="Arg">a</var> and <var class="Arg">b</var>.</p>
<p>The first command computes the product of the left ideal <var class="Arg">a</var> and the right ideal <var class="Arg">b</var>. If they are not appropriately-sided ideals, the command first attempts to convert them.</p>
<p>The second command assumes that the ring of these ideals has a basis made of invertible elements. It is then much easier to compute the product.</p>
<p><a id="X794940277C9AD85C" name="X794940277C9AD85C"></a></p>
<h5>11.8-11 DimensionSeries</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; DimensionSeries</code>( <var class="Arg">a</var>[, <var class="Arg">n</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: A nested list of ideals in the algebra-with-one <var class="Arg">a</var>.</p>
<p>This command computes the powers of the augmentation ideal of <var class="Arg">a</var>, and returns their list. The list stops when the list becomes stationary.</p>
<p>The optional second argument gives a limit to the number of terms to put in the series.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">a := ThinnedAlgebraWithOne(GF(2),GrigorchukGroup);</span>
<self-similar algebra-with-one on alphabet GF(2)^2 with 4 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">q := MatrixQuotient(a,3);</span>
<algebra-with-one of dimension 22 over GF(2)>
<span class="GAPprompt">gap></span> <span class="GAPinput">l := DimensionSeries(q);</span>
[ <two-sided ideal in <algebra-with-one of dimension 22 over GF(2)>, (5 generators)>,
<two-sided ideal in <algebra-with-one of dimension 22 over GF(2)>, (dimension 21)>,
<two-sided ideal in <algebra-with-one of dimension 22 over GF(2)>, (dimension 18)>,
<two-sided ideal in <algebra-with-one of dimension 22 over GF(2)>, (dimension 14)>,
<two-sided ideal in <algebra-with-one of dimension 22 over GF(2)>, (dimension 10)>,
<two-sided ideal in <algebra-with-one of dimension 22 over GF(2)>, (dimension 6)>,
<two-sided ideal in <algebra-with-one of dimension 22 over GF(2)>, (dimension 3)>,
<two-sided ideal in <algebra-with-one of dimension 22 over GF(2)>, (dimension 1)>,
<algebra of dimension 0 over GF(2)> ]
</pre></div>
<p><a id="X825149467C57DEFC" name="X825149467C57DEFC"></a></p>
<h5>11.8-12 AlgebraHomomorphismByFunction</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; AlgebraHomomorphismByFunction</code>( <var class="Arg">A</var>, <var class="Arg">B</var>, <var class="Arg">f</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">an style='color: green'>8227; AlgebraWithOneHomomorphismByFunction</code>( <var class="Arg">A</var>, <var class="Arg">B</var>, <var class="Arg">f</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: A homomorphism from the algebra <var class="Arg">A</var> to the algebra <var class="Arg">B</var>.</p>
<p>These functions construct an algebra homomorphism from a one-argument function. They do not check that the function actually defines a homomorphism.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A := MatrixAlgebra(Rationals,2);</span>
( Rationals^[ 2, 2 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">e1 := AlgebraHomomorphismByFunction(Rationals,A,f->[[f,0],[0,0]]);</span>
MappingByFunction( Rationals, ( Rationals^[ 2, 2 ] ), function( f ) ... end )
<span class="GAPprompt">gap></span> <span class="GAPinput">11^e1;</span>
[ [ 11, 0 ], [ 0, 0 ] ]
</pre></div>
<p><a id="X837F6C2A797FFD41" name="X837F6C2A797FFD41"></a></p>
<h5>11.8-13 IsFpLieAlgebra</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; IsFpLieAlgebra</code></td><td class="tdright">( filter )</td></tr></table></div>
<p>The category of Lie algebras coming from a finitely presented group. They appear as the <code class="func">JenningsLieAlgebra</code> (<a href="../../../doc/ref/chap64_mj.html#X8692ADD581359CA1"><span class="RefLink">Reference: JenningsLieAlgebra</span></a>) of a finitely presented group.</p>
<p>If <code class="code">G</code> is an infinite, finitely presented group, then the original implementation of <code class="func">JenningsLieAlgebra</code> (<a href="../../../doc/ref/chap64_mj.html#X8692ADD581359CA1"><span class="RefLink">Reference: JenningsLieAlgebra</span></a>) does not return. On the other hand, the implementation in <strong class="pkg">FR</strong> constructs a graded object, for which the graded components are computed on-demand; see <code class="func">JenningsLieAlgebra</code> (<a href="chap11_mj.html#X8692ADD581359CA1"><span class="RefLink">11.8-14</span></a>).</p>
<p><a id="X8692ADD581359CA1" name="X8692ADD581359CA1"></a></p>
<h5>11.8-14 JenningsLieAlgebra</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; JenningsLieAlgebra</code>( <var class="Arg">ring</var>, <var class="Arg">fpgroup</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: The Jennings Lie algebra of <var class="Arg">fpgroup</var>.</p>
<p>This method does not compute the Jennings Lie algebra <em>per se</em>; it merely constructs a placeholder to contain the result.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">f := FreeGroup(4);</span>
<free group on the generators [ f1, f2, f3, f4 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">surfacegp := f/[Comm(f.1,f.2)*Comm(f.3,f.4)];</span>
<fp group of size infinity on the generators [ f1, f2, f3, f4 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">j := JenningsLieAlgebra(Rationals,surfgp);</span>
<FP Lie algebra over Rationals>
<span class="GAPprompt">gap></span> <span class="GAPinput">List([1..4],Grading(j).hom_components);</span>
[ <vector space over Rationals, with 4 generators>,
<vector space over Rationals, with 5 generators>,
<vector space over Rationals, with 16 generators>,
<vector space over Rationals, with 45 generators> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">B := Basis(Grading(j).hom_components(1));</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">B[1]*B[2]+B[3]*B[4];</span>
<zero Lie element>
</pre></div>
<p><a id="X7F5427847E43CD62" name="X7F5427847E43CD62"></a></p>
<h5>11.8-15 SolutionMatModN</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; SolutionMatModN</code>( <var class="Arg">mat</var>, <var class="Arg">vec</var>, <var class="Arg">N</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Solve the linear system <code class="code">sol*mat=vec</code> modulo <var class="Arg">N</var>. The arguments are assumed to be an integer matrix and vector. Either returns an integer solution, or <code class="keyw">fail</code> if no such solution exists.</p>
<p><a id="X80E9634283E2DF18" name="X80E9634283E2DF18"></a></p>
<h5>11.8-16 SolutionMatMod1</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; SolutionMatMod1</code>( <var class="Arg">mat</var>, <var class="Arg">vec</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Solve the linear system <code class="code">sol*mat=vec</code> in <span class="SimpleMath">\(Q/Z\)</span>. The arguments are assumed to be rational matrices. Assuming there are finitely many solutions, returns them all.</p>
<p><a id="X7D2CEF7F845D2AF2" name="X7D2CEF7F845D2AF2"></a></p>
<h5>11.8-17 CyclotomicByArgument</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; CyclotomicByArgument</code>( <var class="Arg">q</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: The cyclotomic field element equal to <span class="SimpleMath">\(\exp(2\pi i q)\)</span>.</p>
<p><a id="X7A6CB5FE86501506" name="X7A6CB5FE86501506"></a></p>
<h5>11.8-18 ArgumentOfCyclotomic</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; ArgumentOfCyclotomic</code>( <var class="Arg">z</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: The rational <span class="SimpleMath">\(q\)</span> such that <span class="SimpleMath">\(\exp(2\pi i q)=z\)</span>.</p>
<p><a id="X7ADFF37084706CEC" name="X7ADFF37084706CEC"></a></p>
<h4>11.9 <span class="Heading">User settings</span></h4>
<p><a id="X7E95AC267805E4C1" name="X7E95AC267805E4C1"></a></p>
<h5>11.9-1 InfoFR</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; InfoFR</code></td><td class="tdright">( info class )</td></tr></table></div>
<p>This is an <code class="keyw">Info</code> class for the package <strong class="pkg">FR</strong>. The command <code class="code">SetInfoLevel(InfoFR,1);</code> switches on the printing of some information during the computations of certain <strong class="pkg">FR</strong> functions; in particular all automatic conversions between FR machines and Mealy machines.</p>
<p>The command <code class="code">SetInfoLevel(InfoFR,2);</code> requests a little more information, and in particular prints intermediate results in potentially long calculations such as <code class="func">NucleusOfFRSemigroup</code> (<a href="chap7_mj.html#X7CA062A67C1554BB"><span class="RefLink">7.2-19</span></a>).</p>
<p>The command <code class="code">SetInfoLevel(InfoFR,3);</code> ensures that <strong class="pkg">FR</strong> will print information every few seconds or so. This is useful to gain confidence that the program is not stuck due to a programming bug by the author of <strong class="pkg">FR</strong>.</p>
<p><a id="X835FBD72853595BE" name="X835FBD72853595BE"></a></p>
<h5>11.9-2 SEARCH@</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; SEARCH@</code></td><td class="tdright">( global variable )</td></tr></table></div>
<p>This variable controls the search mechanism in FR groups. It is a record with in particular entries <code class="code">radius</code> and <code class="code">depth</code>.</p>
<p><code class="code">radius</code> limits the search in FR groups to balls of that radius in the generating set. For example, the command <code class="code">x in G</code> will initiate a search in <code class="code">G</code> to attempt to express <code class="code">x</code> as a reasonably short word in the generators of <code class="code">G</code>.</p>
<p><code class="code">depth</code> limits the level of the tree on which quotients of FR groups should be considered. Again for the command <code class="code">x in G</code>, deeper and deeper quotients will be considered, in the hope of finding a quotient of <code class="code">G</code> to which <code class="code">x</code> does not belong.</p>
<p>A primitive mechanism is implemented to search alternatively for a quotient disproving <code class="code">x in G</code> and a word proving <code class="code">x in G</code>.</p>
<p>When the limits are reached and the search was unsuccessful, an interactive <code class="code">Error()</code> is raised, to let the user increase their values.</p>
<p>Specific limits can be passed to any command via the options <code class="code">FRdepth</code> and <code class="code">FRradius</code>, as for example in <code class="code">Size(G:FRdepth:=3,FRradius:=5)</code>.</p>
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