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<div class="ChapSects"><a href="chap48.html#X782985197BE809BF">48 <span class="Heading">Presentations and Tietze Transformations</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap48.html#X867D00387957450F">48.1 <span class="Heading">Creating Presentations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap48.html#X797867B287AD92F8">48.1-1 PresentationFpGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap48.html#X8637837A79422445">48.1-2 TzSort</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap48.html#X849429BC7D435F77">48.1-3 GeneratorsOfPresentation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap48.html#X7D6F40A87F24D3D6">48.1-4 FpGroupPresentation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap48.html#X84E056C57AFEDEA8">48.1-5 PresentationViaCosetTable</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap48.html#X7E1F2658827FC228">48.1-6 SimplifiedFpGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap48.html#X8118FECE7AD1879B">48.2 <span class="Heading">Subgroup Presentations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap48.html#X857365CD87ADC29E">48.2-2 <span class="Heading">PresentationSubgroupRrs</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap48.html#X7FCE7ED581CF7897">48.2-3 PrimaryGeneratorWords</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap48.html#X80BA10F780EAE68E">48.2-4 PresentationSubgroupMtc</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap48.html#X7D6A52837BEE5C3D">48.2-5 PresentationNormalClosureRrs</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap48.html#X7A7E5D0084DB0B4F">48.2-6 PresentationNormalClosure</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap48.html#X7BC960AB7E8DE419">48.3 <span class="Heading">Relators in a Presentation</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap48.html#X8365BAFA785FCD8D">48.3-1 TietzeWordAbstractWord</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap48.html#X8573E91C838B1D13">48.3-2 AbstractWordTietzeWord</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap48.html#X867F64FA840B3F81">48.4 <span class="Heading">Printing Presentations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap48.html#X847EA6737C21171C">48.4-1 TzPrintGenerators</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap48.html#X821B63DD82894443">48.4-2 TzPrintRelators</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap48.html#X852C52C37FAAB7DD">48.4-3 TzPrintLengths</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap48.html#X7D7B3F46865443E4">48.4-4 TzPrintStatus</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap48.html#X85F8DAE27F06C32B">48.4-5 TzPrintPresentation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap48.html#X7CA8BA51802655FC">48.4-6 TzPrint</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap48.html#X82F6B0EE7C7C7901">48.4-7 TzPrintPairs</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap48.html#X82455E5885D73FFF">48.5 <span class="Heading">Changing Presentations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap48.html#X7F632A6D8685855D">48.5-1 AddGenerator</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap48.html#X83A5667086FD538A">48.5-2 TzNewGenerator</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap48.html#X78D1BCE67FA852D8">48.5-3 AddRelator</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap48.html#X7B11E89E78A22EBF">48.5-4 RemoveRelator</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap48.html#X829B634286471AB7">48.6 <span class="Heading">Tietze Transformations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap48.html#X7C4A30328224C466">48.6-1 TzGo</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap48.html#X78C3D23387DAC35A">48.6-2 SimplifyPresentation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap48.html#X801D3D8984E1CA55">48.6-3 TzGoGo</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap48.html#X7D19E30080290FC7">48.7 <span class="Heading">Elementary Tietze Transformations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap48.html#X85989AF886EC2BF6">48.7-1 <span class="Heading">TzEliminate</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap48.html#X7DF4BBDF839643DD">48.7-2 TzSearch</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap48.html#X87F7A87A7ACF2445">48.7-3 TzSearchEqual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap48.html#X80D31A0F7C2A51BD">48.7-4 TzFindCyclicJoins</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap48.html#X7D2FACCF79F57040">48.8 <span class="Heading">Tietze Transformations that introduce new Generators</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap48.html#X846DB23E8236FF8A">48.8-1 <span class="Heading">TzSubstitute</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap48.html#X7ADE3B437C19B94D">48.8-2 TzSubstituteCyclicJoins</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap48.html#X85E703997A0212EE">48.9 <span class="Heading">Tracing generator images through Tietze transformations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap48.html#X7D855FA08242898A">48.9-1 TzInitGeneratorImages</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap48.html#X7AB9A06F80FB3659">48.9-2 OldGeneratorsOfPresentation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap48.html#X798B38F87C082C45">48.9-3 TzImagesOldGens</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap48.html#X7AC41B117DBB87D6">48.9-4 TzPreImagesNewGens</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap48.html#X7F086D0E7AD6173B">48.9-5 TzPrintGeneratorImages</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap48.html#X7D9E283D81CCCF1A">48.10 <span class="Heading">The Decoding Tree Procedure</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap48.html#X7ACBFE2F78D72A31">48.10-1 DecodeTree</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap48.html#X856F37537E9927EE">48.11 <span class="Heading">Tietze Options</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap48.html#X8178683283214D88">48.11-1 TzOptions</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap48.html#X7BC90B6882DE6D10">48.11-2 TzPrintOptions</a></span>
</div></div>
</div>
<h3>48 <span class="Heading">Presentations and Tietze Transformations</span></h3>
<p>A finite presentation describes a group, but usually there is a multitude of presentations that describe isomorphic groups. Therefore a presentation in <strong class="pkg">GAP</strong> is different from a finitely presented group though there are ways to translate between both.</p>
<p>An important feature of presentations is that they can be modified (see sections <a href="chap48.html#X82455E5885D73FFF"><span class="RefLink">48.5</span></a> to <a href="chap48.html#X7D2FACCF79F57040"><span class="RefLink">48.8</span></a>).</p>
<p>If you only want to get new presentations for subgroups of a finitely presented group (and do not want to manipulate presentations yourself), chances are that the operation <code class="func">IsomorphismFpGroup</code> (<a href="chap47.html#X7F28268F850F454E"><span class="RefLink">47.11-1</span></a>) already does what you want (see <a href="chap47.html#X826604AA7F18BFA3"><span class="RefLink">47.12</span></a>).</p>
<p><a id="X867D00387957450F" name="X867D00387957450F"></a></p>
<h4>48.1 <span class="Heading">Creating Presentations</span></h4>
<p>Most of the functions creating presentations and all functions performing Tietze transformations on them sort the relators by increasing lengths. The function <code class="func">PresentationFpGroup</code> (<a href="chap48.html#X797867B287AD92F8"><span class="RefLink">48.1-1</span></a>) is an exception because it is intended to reflect the relators that were used to define the involved f. p. group. You may use the command <code class="func">TzSort</code> (<a href="chap48.html#X8637837A79422445"><span class="RefLink">48.1-2</span></a>) to sort the presentation.</p>
<p><a id="X797867B287AD92F8" name="X797867B287AD92F8"></a></p>
<h5>48.1-1 PresentationFpGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; PresentationFpGroup</code>( <var class="Arg">G</var>[, <var class="Arg">printlevel</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>creates a presentation, i. e., a Tietze object, for the given finitely presented group <var class="Arg">G</var>. This presentation will be exactly as the presentation of <var class="Arg">G</var> and <em>no</em> initial Tietze transformations are applied to it.</p>
<p>The optional <var class="Arg">printlevel</var> parameter can be used to restrict or to extend the amount of output provided by Tietze transformation commands when being applied to the created presentation. The default value 1 is designed for interactive use and implies explicit messages to be displayed by most of these commands. A <var class="Arg">printlevel</var> value of 0 will suppress these messages, whereas a <var class="Arg">printlevel</var> value of 2 will enforce some additional output.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">f := FreeGroup( "a", "b" );</span>
<free group on the generators [ a, b ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">g := f / [ f.1^3, f.2^2, (f.1*f.2)^3 ];</span>
<fp group on the generators [ a, b ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">p := PresentationFpGroup( g );</span>
<presentation with 2 gens and 3 rels of total length 11>
</pre></div>
<p><a id="X8637837A79422445" name="X8637837A79422445"></a></p>
<h5>48.1-2 TzSort</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; TzSort</code>( <var class="Arg">P</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>sorts the relators of the given presentation <var class="Arg">P</var> by increasing lengths. There is no particular ordering defined for the relators of equal length. Note that <code class="func">TzSort</code> does not return a new object. It changes the given presentation.</p>
<p><a id="X849429BC7D435F77" name="X849429BC7D435F77"></a></p>
<h5>48.1-3 GeneratorsOfPresentation</h5>
<div class="func"<table class=func width=100%><><td"P/var)/td<td =tdright">(nbsp;function&bsp)/td</tr><table>/div
<p>returns a list of free generators that is a shallow copy (see <code class="func">ShallowCopy</code>java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
<p><a id="X7D6F40A87F24D3D6" name="X7D6F40A87F24D3D6"></a></p>
<h5>48.1-4 FpGroupPresentation</h5>
<div class="func"><table class="func" width="100%"><tr><td <iv class="">table =" ="00%<tr< class=dleft"< =""#;IsQuasiPrimitive/code>(var =Arg"><var> </td< java.lang.StringIndexOutOfBoundsException: Range [169, 168) out of bounds for length 225
<constructs an f. p. group as defined by the given Tietze presentation <var class="Arg">P</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">h := FpGroupPresentation( p );</span>
p< =func>IsQuasiPrimitive<code <code ""truecode>or < class=keyw"false/code,depending java.lang.StringIndexOutOfBoundsException: Range [134, 133) out of bounds for length 205
<span class="GAPprompt">gap></span> <span class="GAPinput">h java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
false
</pre></div>
<p><a id="X84E056C57AFEDEA8" name="X84E056C57AFEDEA8"></a></p>
<h5>48.1-5 <span class="GAPprompt">gap>span><panclass="GAPinput">chi:= Irr( Sl23 )[4];</span>
<class=func>tableclass="func" width="100">tr><tdclass="tdleft"< ="func">‣ PresentationViaCosetTable</code>( <var class="Arg">G</var>[, <var class="Arg">F<class="GAPprompt">gap></span> <span class="GAPinput">TestQuasiPrimitive( chi );</span>
<p>constructs a presentation for a given concrete finite group. Itjava.lang.StringIndexOutOfBoundsException: Range [10, 9) out of bounds for length 64
<p>If only a group <var class="Arg">G</varspan class="APprompt"<span> <class=GAPinput">egree( java.lang.StringIndexOutOfBoundsException: Range [75, 74) out of bounds for length 94
<>The operation code class=""IsomorphismFpGroup</code> (<a href="chap47.html#X7F28268F850F454E"><span class="RefLink">47.11-1</span></a>) in contrast uses a multiple-stage algorithm using a chief series and stabilizer chains. It usually should be used rather than <code class="func">PresentationViaCosetTable</code>. (It does not apply Tietze java.lang.StringIndexOutOfBoundsException: Index 354 out of bounds for length 0
<p>If the two stage algorithm is to be used, <code class="func">PresentationViaCosetTable</code> expects a subgroup <var class="Arg">H</var> of <var class="Arg">G</var> to be provided in form of two additional arguments <"<tdjava.lang.StringIndexOutOfBoundsException: Range [174, 173) out of bounds for length 236
<div class="example"><pre>
<spanclass"GAPprompt">&;/pan span=GAPinput"G: (2,7)/>
GL(2,7)
<span class="GAPprompt">gap>
[ p<code ""java.lang.StringIndexOutOfBoundsException: Range [50, 49) out of bounds for length 255
[ [ Z(7)^3, Z(7)^0 ], [ Z(7)^3, 0*Z(7) ] ] ]
<span class="GAPprompt<iv class"xample"><re
2016
<span class= ,false false ,, java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
<presentation with 2 gens and 5 rels of total length 46>< =""&;/><span class="APinput> );/spanjava.lang.StringIndexOutOfBoundsException: Index 107 out of bounds for length 107
<span class="GAPprompt">gap&java.lang.StringIndexOutOfBoundsException: Range [8, 1) out of bounds for length 33
#I 1. f2^3
#I 2. f1^6
#I java.lang.StringIndexOutOfBoundsException: Range [16, 15) out of bounds for length 26
#I 4. f1*f2*f1^h4>75. span ="Testing Monomiality<span></
#I 5
</pre></div>
<p>The two stage
<div class="example"><pre>
<span class
<span class="GAPprompt">></span> <span class=<h5>75.4- <pan class="Heading">TestMonomial</span></h5>
<span classf width"<codeclass="func">‣ TestMonomial</code>( <var class="Arg">chi</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
&erators [ ,b,c ]gt;
<span class="GAPprompt">gap></span> <span div class="func"><table class="func" width="100%"class=""<java.lang.StringIndexOutOfBoundsException: Range [85, 79) out of bounds for length 255
[ a, b ]
<span class="GAPprompt">gap></span> <span< a ""G<>, < class" recordcontaining about monomialityjava.lang.StringIndexOutOfBoundsException: Range [197, 196) out of bounds for length 300
<presentation with 3 gens and 10 rels of total length 97>
spanclass="GAPprompt">gap></span> <span class="GAPinput">G := FpGroupPresentation( P );</span>
<fp p>java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 407
<span class="GAPprompt">gap></span> <span class="GAPinput">RelatorsOfFpGroup( G );</span>
[ c^2, b^4, (a*c)^3, (a*b^-2)^3, a^11,
a^2*b*a^-2*b^-1*(b^-1*a)^2*a*b^-1, (a*(b*
a2*b*^*^2a^-*b*^-1b)2,
a^lattice must not be used, it may happen that <code class="func">TestMonomial</code> cannot prove or disprove monomiality; then the result record contains
]
</pre></div>
<>Before is returned, the resulting presentation is being simplified by appropriate calls of the function <code class="func">SimplifyPresentation</code> (<a href="chap48.html#X78C3D23387DAC35A"><span )
<div class="example"><pre>rec( : "list ( )contains entry > 1"",
<span class="GAPprompt">gap></span> <span class="GAPinput">H := Group(</span>
<span class="GAPprompt">></span> <span class="GAPinput">[ (2,5,3), (2,7,5), (1,8,4),
< H );</span
<presentation with 6 gens and 12 rels of total length 42>
<span class="GAPprompt">gap></span> <span class="GAPinput">SimplifyPresentation( java.lang.StringIndexOutOfBoundsException: Index 85 out of bounds for length 0
#I there are 4 generators and 10 relators of total length 36
</pre></div>
<p>If you apply the function <code class="func">FpGroupPresentationp>For apositive integer <var class="Arg">n</var>, <code class="func">IsMonomialNumber</code> returns <code class="keyw">true</code> if every solvable group of order <java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<"></a></p>
<h5>48.1-6 SimplifiedFpGroup</h5>
<div class="unc>< =
<p>applies Tietze transformations to a copy of the presentation of the given finitely <dd><p><span class="SimpleMath">ν_2(n) ≥ 2</span> and there is a <span class="SimpleMath">p</sp
<p><code class="func">SimplifiedFpGroup</code> returns a group isomorphic to the given one with a presentation which has been tried to simplify via Tietze transformations.</p>
<p>If the connection to the original group is important, then the operation <code class="func">IsomorphismSimplifiedFpGroup</code> (<a href="chap47.html#X78D87FA68233C401"><span class="RefLink">47.12-1</span></a>) should be used instead.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">F6 := FreeGroup( 6, "G" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := F6 / [ F6.1^2, F6.2^2, F6.4*F6.6^-1, F6.5^2, F6.6^2,</span>
<span class="GAPprompt">></span> <span class="GAPinput">F6.1*F6.2^-1*F6.3, F6.1*F6.5*F6.3^-1, F6.2*F6.4^-1*F6.3,</span>
<span class="GAPprompt">></span> <span class="GAPinput">F6.3*F6.4*F6.5^-1, F6.1*F6.6*F6.3^-2, F6.3^4 ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">H := SimplifiedFpGroup( G );</span>
<fp group on the generators [ G1, G3 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">RelatorsOfFpGroup( H );</span>
[ G1^2, (G1*G3^-1)^2, G3^4 ]
</pre></div>
<p>In fact, the command</p>
<div class="example"><pre>
H := SimplifiedFpGroup( G );
</pre></div>
<p>is an abbreviation of the command sequence</p>
<div class="example"><pre>
P := PresentationFpGroup( G, 0 );;
SimplifyPresentation( P );
H := FpGroupPresentation( P );
</pre></div>
<p>which applies a rather simple-minded strategy of Tietze transformations to the intermediate presentation <var class="Arg">P</var>. If, for some concrete group, the resulting presentation is unsatisfying, then you should try a more sophisticated, interactive use of the available Tietze transformation commands (see <a href="chap48.html#X829B634286471AB7"><span class="RefLink">48.6</span></a>).</p>
<p><a id="X8118FECE7AD1879B" name="X8118FECE7AD1879B"></a></p>
<h4>48.2 <span class="Heading">Subgroup Presentations</span></h4>
<p><a id="X7DB32FA97DAC5AC8" name="X7DB32FA97DAC5AC8"></a></p>
<h5>48.2-1 PresentationSubgroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; PresentationSubgroup</code>( <var class="Arg">G</var>, <var class="Arg">H</var>[, <var class="Arg">string</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">PresentationSubgroup</code> is a synonym for <code class="func">PresentationSubgroupRrs</code> (<a href="chap48.html#X857365CD87ADC29E"><span class="RefLink">48.2-2</span></a>).</p>
<p><a id="X857365CD87ADC29E" name="X857365CD87ADC29E"></a></p>
<h5>48.2-2 <span class="Heading">PresentationSubgroupRrs</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; PresentationSubgroupRrs</code>( <var class="Arg">G</var>, <var class="Arg">H</var>[, <var class="Arg">string</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; PresentationSubgroupRrs</code>( <var class="Arg">G</var>, <var class="Arg">table</var>[, <var class="Arg">string</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>uses the Reduced Reidemeister-Schreier method to compute a presentation <var class="Arg">P</var> for a subgroup <var class="Arg">H</var> of a finitely presented group <var class="Arg">G</var>. The generators in the resulting presentation will be named <var class="Arg">string</var><code class="code">1</code>, <var class="Arg">string</var><code class="code">2</code>, <span class="SimpleMath">...</span>, the default string is <code class="code">"_x"</code>. You may access the <span class="SimpleMath">i</span>-th of these generators by <var class="Arg">P</var><code class="code">!.</code><span class="SimpleMath">i</span>.</p>
<p>Alternatively to the subgroup <var class="Arg">H</var>, its coset table <var class="Arg">table</var> in <var class="Arg">G</var> may be given as second argument.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">f := FreeGroup( "a", "b" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">g := f / [ f.1^2, f.2^3, (f.1*f.2)^5 ];</span>
<fp group on the generators [ a, b ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">g1 := Size( g );</span>
60
<span class="GAPprompt">gap></span> <span class="GAPinput">u := Subgroup( g, [ g.1, g.1^g.2 ] );</span>
Group([ a, b^-1*a*b ])
<span class="GAPprompt">gap></span> <span class="GAPinput">p := PresentationSubgroup( g, u, "g" );</span>
<presentation with 3 gens and 4 rels of total length 12>
<span class="GAPprompt">gap></span> <span class="GAPinput">gens := GeneratorsOfPresentation( p );</span>
[ g1, g2, g3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">TzPrintRelators( p );</span>
#I 1. g1^2
#I 2. g2^2
#I 3. g3*g2*g1
#I 4. g3^5
</pre></div>
<p>Note that you cannot call the generators by their names. These names are not variables, but just display figures. So, if you want to access the generators by their names, you first will have to introduce the respective variables and to assign the generators to them.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">gens[1] = g1;</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">g1;</span>
60
<span class="GAPprompt">gap></span> <span class="GAPinput">g1 := gens[1];; g2 := gens[2];; g3 := gens[3];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">g1;</span>
g1
</pre></div>
<p>The Reduced Reidemeister-Schreier algorithm is a modification of the Reidemeister-Schreier algorithm of George Havas <a href="chapBib.html#biBHav74b">[Hav74]</a>. It was proposed by Joachim Neubüser and first implemented in 1986 by Andrea Lucchini and Volkmar Felsch in the SPAS system <a href="chapBib.html#biBSpa89">[SPA89]</a>. Like the Reidemeister-Schreier algorithm of George Havas, it needs only the presentation of <var class="Arg">G</var> and a coset table of <var class="Arg">H</var> in <var class="Arg">G</var> to construct a presentation of <var class="Arg">H</var>.</p>
<p>Whenever you call the command <code class="func">PresentationSubgroupRrs</code>, it first obtains a coset table of <var class="Arg">H</var> in <var class="Arg">G</var> if not given. Next, a set of generators of <var class="Arg">H</var> is determined by reconstructing the coset table and introducing in that process as many Schreier generators of <var class="Arg">H</var> in <var class="Arg">G</var> as are needed to do a Felsch strategy coset enumeration without any coincidences. (In general, though containing redundant generators, this set will be much smaller than the set of all Schreier generators. That is why we call the method the <em>Reduced</em> Reidemeister-Schreier.)</p>
<p>After having constructed this set of <em>primary subgroup generators</em>, say, the coset table is extended to an <em>augmented coset table</em> which describes the action of the group generators on coset representatives, i.e., on elements instead of cosets. For this purpose, suitable words in the (primary) subgroup generators have to be associated to the coset table entries. In order to keep the lengths of these words short, additional <em>secondary subgroup generators</em> are introduced as abbreviations of subwords. Their number may be large.</p>
<p>Finally, a Reidemeister rewriting process is used to get defining relators for <var class="Arg">H</var> from the relators of <var class="Arg">G</var>. As the resulting presentation of <var class="Arg">H</var> is a presentation on primary <em>and</em> secondary generators, in general you will have to simplify it by appropriate Tietze transformations (see <a href="chap48.html#X829B634286471AB7"><span class="RefLink">48.6</span></a>) or by the command <code class="func">DecodeTree</code> (<a href="chap48.html#X7ACBFE2F78D72A31"><span class="RefLink">48.10-1</span></a>) before you can use it. Therefore it is returned in the form of a presentation, <var class="Arg">P</var> say.</p>
<p>Compared with the Modified Todd-Coxeter method described below, the Reduced Reidemeister-Schreier method (as well as Havas' original Reidemeister-Schreier program) has the advantage that it does not require generators of <var class="Arg">H</var> to be given if a coset table of <var class="Arg">H</var> in <var class="Arg">G</var> is known. This provides a possibility to compute a presentation of the normal closure of a given subgroup (see <code class="func">PresentationNormalClosureRrs</code> (<a href="chap48.html#X7D6A52837BEE5C3D"><span class="RefLink">48.2-5</span></a>)).</p>
<p>For certain applications you may be interested in getting not only just a presentation for <var class="Arg">H</var>, but also a relation between the involved generators of <var class="Arg">H</var> and the generators of <var class="Arg">G</var>. The subgroup generators in the presentation are sorted such that the primary generators precede the secondary ones. Moreover, for each secondary subgroup generator there is a relator in the presentation which expresses this generator as a word in preceding ones. Hence, all we need in addition is a list of words in the generators of <var class="Arg">G</var> which express the primary subgroup generators. In fact, such a list is provided in the attribute <code class="func">PrimaryGeneratorWords</code> (<a href="chap48.html#X7FCE7ED581CF7897"><span class="RefLink">48.2-3</span></a>) of the resulting presentation.</p>
<p><a id="X7FCE7ED581CF7897" name="X7FCE7ED581CF7897"></a></p>
<h5>48.2-3 PrimaryGeneratorWords</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; PrimaryGeneratorWords</code>( <var class="Arg">P</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>is an attribute of the presentation <var class="Arg">P</var> which holds a list of words in the associated group generators (of the underlying free group) which express the primary subgroup generators of <var class="Arg">P</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">PrimaryGeneratorWords( p );</span>
[ a, b^-1*a*b ]
</pre></div>
<p><a id="X80BA10F780EAE68E" name="X80BA10F780EAE68E"></a></p>
<h5>48.2-4 PresentationSubgroupMtc</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; PresentationSubgroupMtc</code>( <var class="Arg">G</var>, <var class="Arg">H</var>[, <var class="Arg">string</var>][, <var class="Arg">print</var>, <var class="Arg">level</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>uses the Modified Todd-Coxeter coset representative enumeration method to compute a presentation <span class="SimpleMath">P</span> for a subgroup <var class="Arg">H</var> of a finitely presented group <var class="Arg">G</var>. The presentation returned is in generators corresponding to the generators of <var class="Arg">H</var>. The generators in the resulting presentation will be named <var class="Arg">string</var><code class="code">1</code>, <var class="Arg">string</var><code class="code">2</code>, <span class="SimpleMath">...</span>, the default string is <code class="code">"_x"</code>. You may access the <span class="SimpleMath">i</span>-th of these generators by <span class="SimpleMath">P</span><code class="code">!.</code><span class="SimpleMath">i</span>.</p>
<p>The default print level is <span class="SimpleMath">1</span>. If the print level is set to <span class="SimpleMath">0</span>, then the printout of the implicitly called function <code class="func">DecodeTree</code> (<a href="chap48.html#X7ACBFE2F78D72A31"><span class="RefLink">48.10-1</span></a>) will be suppressed.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">p := PresentationSubgroupMtc( g, u );</span>
<presentation with 2 gens and 3 rels of total length 14>
</pre></div>
<p>The so called Modified Todd-Coxeter method was proposed, in slightly different forms, by Nathan S. Mendelsohn and William O. J. Moser in 1966. Moser's method was proved in <a href="chapBib.html#biBBC76">[BC76]</a>. It has been generalized to cover a broad spectrum of different versions (see the survey <a href="chapBib.html#biBNeu82">[Neu82]</a>).</p>
<p>The <em>Modified Todd-Coxeter</em> method performs an enumeration of coset representatives. It proceeds like an ordinary coset enumeration (see <a href="chap47.html#X7BD0CEBA7B225416"><span class="RefLink">47.6</span></a>), but as the product of a coset representative by a group generator or its inverse need not be a coset representative itself, the Modified Todd-Coxeter has to store a kind of correction element for each coset table entry. Hence it builds up a so called <em>augmented coset table</em> of <var class="Arg">H</var> in <var class="Arg">G</var> consisting of the ordinary coset table and a second table in parallel which contains the associated subgroup elements.</p>
<p>Theoretically, these subgroup elements could be expressed as words in the given generators of <var class="Arg">H</var>, but in general these words tend to become unmanageable because of their enormous lengths. Therefore, a highly redundant list of subgroup generators is built up starting from the given (<q>primary</q>) generators of <var class="Arg">H</var> and adding additional (<q>secondary</q>) generators which are defined as abbreviations of suitable words of length two in the preceding generators such that each of the subgroup elements in the augmented coset table can be expressed as a word of length at most one in the resulting (primary <em>and</em> secondary) subgroup generators.</p>
<p>Then a rewriting process (which is essentially a kind of Reidemeister rewriting process) is used to get relators for <var class="Arg">H</var> from the defining relators of <var class="Arg">G</var>.</p>
<p>The resulting presentation involves all the primary, but not all the secondary generators of <var class="Arg">H</var>. In fact, it contains only those secondary generators which explicitly occur in the augmented coset table. If we extended this presentation by those secondary generators which are not yet contained in it as additional generators, and by the definitions of all secondary generators as additional relators, we would get a presentation of <var class="Arg">H</var>, but, in general, we would end up with a large number of generators and relators.</p>
<p>On the other hand, if we avoid this extension, the current presentation will not necessarily define <var class="Arg">H</var> although we have used the same rewriting process which in the case of the <code class="func">PresentationSubgroupRrs</code> (<a href="chap48.html#X857365CD87ADC29E"><span class="RefLink">48.2-2</span></a>) command computes a defining set of relators for <var class="Arg">H</var> from an augmented coset table and defining relators of <var class="Arg">G</var>. The different behaviour here is caused by the fact that coincidences may have occurred in the Modified Todd-Coxeter coset enumeration.</p>
<p>To overcome this problem without extending the presentation by all secondary generators, the <code class="func">PresentationSubgroupMtc</code> command applies the so called <em>decoding tree</em> algorithm which provides a more economical approach. The reader is strongly recommended to carefully read section <a href="chap48.html#X7D9E283D81CCCF1A"><span class="RefLink">48.10</span></a> where this algorithm is described in more detail. Here we will only mention that this procedure may add a lot of intermediate generators and relators (and even change the isomorphism type) in a process which in fact eliminates all secondary generators from the presentation and hence finally provides a presentation of <var class="Arg">H</var> on the primary, i.e., the originally given, generators of <var class="Arg">H</var>. This is a remarkable advantage of the command <code class="func">PresentationSubgroupMtc</code> compared to the command <code class="func">PresentationSubgroupRrs</code> (<a href="chap48.html#X857365CD87ADC29E"><span class="RefLink">48.2-2</span></a>). But note that, for some particular subgroup <var class="Arg">H</var>, the Reduced Reidemeister-Schreier method might quite well produce a more concise presentation.</p>
<p>The resulting presentation is returned in the form of a presentation, <span class="SimpleMath">P</span> say.</p>
<p>As the function <code class="func">PresentationSubgroupRrs</code> (<a href="chap48.html#X857365CD87ADC29E"><span class="RefLink">48.2-2</span></a>) described above (see there for details), the function <code class="func">PresentationSubgroupMtc</code> returns a list of the primary subgroup generators of <var class="Arg">H</var> in the attribute <code class="func">PrimaryGeneratorWords</code> (<a href="chap48.html#X7FCE7ED581CF7897"><span class="RefLink">48.2-3</span></a>) of <span class="SimpleMath">P</span>. In fact, this list is not very exciting here because it is just a shallow copy of the value of <code class="func">GeneratorsOfPresentation</code> (<a href="chap48.html#X849429BC7D435F77"><span class="RefLink">48.1-3</span></a>) of <var class="Arg">H</var>, however it is needed to guarantee a certain consistency between the results of the different functions for computing subgroup presentations.</p>
<p>Though the decoding tree routine already involves a lot of Tietze transformations, we recommend that you try to further simplify the resulting presentation by appropriate Tietze transformations (see <a href="chap48.html#X829B634286471AB7"><span class="RefLink">48.6</span></a>).</p>
<p><a id="X7D6A52837BEE5C3D" name="X7D6A52837BEE5C3D"></a></p>
<h5>48.2-5 PresentationNormalClosureRrs</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; PresentationNormalClosureRrs</code>( <var class="Arg">G</var>, <var class="Arg">H</var>[, <var class="Arg">string</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>uses the Reduced Reidemeister-Schreier method to compute a presentation <span class="SimpleMath">P</span> for the normal closure of a subgroup <var class="Arg">H</var> of a finitely presented group <var class="Arg">G</var>. The generators in the resulting presentation will be named <var class="Arg">string</var><code class="code">1</code>, <var class="Arg">string</var><code class="code">2</code>, <span class="SimpleMath">...</span>, the default string is <code class="code">"_x"</code>. You may access the <span class="SimpleMath">i</span>-th of these generators by <span class="SimpleMath">P</span><code class="code">!.</code><span class="SimpleMath">i</span>.</p>
<p><a id="X7A7E5D0084DB0B4F" name="X7A7E5D0084DB0B4F"></a></p>
<h5>48.2-6 PresentationNormalClosure</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; PresentationNormalClosure</code>( <var class="Arg">G</var>, <var class="Arg">H</var>[, <var class="Arg">string</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">PresentationNormalClosure</code> is a synonym for <code class="func">PresentationNormalClosureRrs</code> (<a href="chap48.html#X7D6A52837BEE5C3D"><span class="RefLink">48.2-5</span></a>).</p>
<p><a id="X7BC960AB7E8DE419" name="X7BC960AB7E8DE419"></a></p>
<h4>48.3 <span class="Heading">Relators in a Presentation</span></h4>
<p>In order to speed up the Tietze transformation routines, each relator in a presentation is internally represented by a list of positive or negative generator numbers, i.e., each factor of the proper <strong class="pkg">GAP</strong> word is represented by the position number of the corresponding generator with respect to the current list of generators, or by the respective negative number, if the factor is the inverse of a generator. Note that the numbering of the generators in Tietze words is always relative to a generator list and bears no relation to the internal numbering of generators in a family of associative words.</p>
<p><a id="X8365BAFA785FCD8D" name="X8365BAFA785FCD8D"></a></p>
<h5>48.3-1 TietzeWordAbstractWord</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; TietzeWordAbstractWord</code>( <var class="Arg">word</var>, <var class="Arg">fgens</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>assumes <var class="Arg">fgens</var> to be a list of free group generators and <var class="Arg">word</var> to be an abstract word in these generators. It converts <var class="Arg">word</var> into a Tietze word, i. e., a list of positive or negative generator numbers.</p>
<p>This function simply calls <code class="func">LetterRepAssocWord</code> (<a href="chap37.html#X7BD7647C7B088389"><span class="RefLink">37.6-8</span></a>).</p>
<p><a id="X8573E91C838B1D13" name="X8573E91C838B1D13"></a></p>
<h5>48.3-2 AbstractWordTietzeWord</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; AbstractWordTietzeWord</code>( <var class="Arg">word</var>, <var class="Arg">fgens</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>assumes <var class="Arg">fgens</var> to be a list of free group generators and <var class="Arg">word</var> to be a Tietze word in these generators, i. e., a list of positive or negative generator numbers. It converts <var class="Arg">word</var> to an abstract word.</p>
<p>This function simply calls <code class="func">AssocWordByLetterRep</code> (<a href="chap37.html#X7AC8EC757CFB9A51"><span class="RefLink">37.6-9</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">F := FreeGroup( "a", "b", "c" ,"d");</span>
<free group on the generators [ a, b, c, d ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">tzword := TietzeWordAbstractWord(</span>
<span class="GAPprompt">></span> <span class="GAPinput">Comm(F.4,F.2) * (F.3^2 * F.2)^-1, GeneratorsOfGroup( F ){[2,3,4]} );</span>
[ -3, -1, 3, -2, -2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">AbstractWordTietzeWord( tzword, GeneratorsOfGroup( F ){[2,3,4]} );</span>
d^-1*b^-1*d*c^-2
</pre></div>
<p><a id="X867F64FA840B3F81" name="X867F64FA840B3F81"></a></p>
<h4>48.4 <span class="Heading">Printing Presentations</span></h4>
<p>Whenever you create a presentation <span class="SimpleMath">P</span>, or assign it to a variable, <strong class="pkg">GAP</strong> will respond by printing <span class="SimpleMath">P</span>. However, as <span class="SimpleMath">P</span> may contain a lot of generators and many relators of large length, it would be annoying if the standard print facilities displayed all this information in detail. So they restrict the printout to just one line of text containing the number of generators, the number of relators, and the total length of all relators of <span class="SimpleMath">P</span>. As compensation, <strong class="pkg">GAP</strong> offers some special print commands which display various details of a presentation. Note that there is also a function <code class="func">TzPrintOptions</code> (<a href="chap48.html#X7BC90B6882DE6D10"><span class="RefLink">48.11-2</span></a>). It is described in Section <a href="chap48.html#X856F37537E9927EE"><span class="RefLink">48.11</span></a>.</p>
<p><a id="X847EA6737C21171C" name="X847EA6737C21171C"></a></p>
<h5>48.4-1 TzPrintGenerators</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; TzPrintGenerators</code>( <var class="Arg">P</var>[, <var class="Arg">list</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>prints the generators of the given Tietze presentation <var class="Arg">P</var> together with the number of their occurrences in the relators. The optional second argument can be used to specify the numbers of the generators to be printed. Default: all generators are printed.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := Group( [ (1,2,3,4,5), (2,3,5,4), (1,6)(3,4) ], () );</span>
Group([ (1,2,3,4,5), (2,3,5,4), (1,6)(3,4) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">P := PresentationViaCosetTable( G );</span>
<presentation with 3 gens and 6 rels of total length 28>
<span class="GAPprompt">gap></span> <span class="GAPinput">TzPrintGenerators( P );</span>
#I 1. f1 11 occurrences
#I 2. f2 10 occurrences
#I 3. f3 7 occurrences involution
</pre></div>
<p><a id="X821B63DD82894443" name="X821B63DD82894443"></a></p>
<h5>48.4-2 TzPrintRelators</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; TzPrintRelators</code>( <var class="Arg">P</var>[, <var class="Arg">list</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>prints the relators of the given Tietze presentation <var class="Arg">P</var>. The optional second argument <var class="Arg">list</var> can be used to specify the numbers of the relators to be printed. Default: all relators are printed.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">TzPrintRelators( P );</span>
#I 1. f3^2
#I 2. f2^4
#I 3. (f2*f3)^2
#I 4. f1^5
#I 5. f1^2*f2*f1*f2^-1
#I 6. f1*f2^-2*f3*f1*f3*f1^-1*f3
</pre></div>
<p><a id="X852C52C37FAAB7DD" name="X852C52C37FAAB7DD"></a></p>
<h5>48.4-3 TzPrintLengths</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; TzPrintLengths</code>( <var class="Arg">P</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>prints just a list of all relator lengths of the given presentation <var class="Arg">P</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">TzPrintLengths( P );</span>
[ 2, 4, 4, 5, 5, 8 ]
</pre></div>
<p><a id="X7D7B3F46865443E4" name="X7D7B3F46865443E4"></a></p>
<h5>48.4-4 TzPrintStatus</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; TzPrintStatus</code>( <var class="Arg">P</var>[, <var class="Arg">norepeat</var>] )</td><td class="tdright">( function )</td></tr></table></div> | |
