/>
<div class="func"><table class="func" width="100%"><tr><tdspan class>gt;> < classGAPinputPrint: "r"( ,(r),)n)<span
<p>returns a finite L-presentation for the lamplighter group on <var class<panclass"GAPprompt>> >pre
<span="GAPprompt">gap;</panspan="GAPinput">LamplighterGroupIsLpGroup );</span
<L-presented group on the ](ollected 0 8]]
<span class=:dquotient invariants,0 0,0 ,0 0 , 0 0,0 ,0 0, 0 , 0 ,0 ,0
&;L-presented onthe [ a ,u ]gt
</pre></div>

<p><a id="X7DBA63A37853BE46" name="X7DBA63A37853BE46"></a></p>

<h5>2.2-8 EmbeddingOfIASubgroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EmbeddingOfIASubgroup</code>( <var
<p>computes

<p>The L-presentation is taken from

<div class="example"><pre>
< class=GAPpromptgap;/span<span=GAPinputf =FreeGroup)<span
<free group on the generators
<span class="GAPprompt">gap>div"

Quellcode-Bibliothek chap2.html Sprache: HTML

products/sources/formale Sprachen/GAP/pkg/lpres/doc/chap2.html

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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X81065E797A486D0F">2.2 <span class="Heading">Creating an <pan=< > class; ><a ="chap2htmlX7F883CC57A3CCAC7">231FreeGroupOfLpGroupa<span
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<span class="ContSS">span=""><br> class&;&;</><a=chap2java.lang.StringIndexOutOfBoundsException: Range [104, 103) out of bounds for length 138
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X81C3537083E40A5C">2.2-4 FreeBurnsideGroup</a></span>
<span class="ContSS"><br /><span class="nocss"/>
span="ContSS">/>span">  /span>2.2-6GeneralizedFabrykowskiGuptaLpGroup/pan
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X80B65AF48662DE70">2.3 <span class="Heading">The underlying free group</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">nbspnbsp<span =htmlX7F883CC57A3CCAC7">2.-
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X838079A587E8CF43">2.3-2 FreeGeneratorsOfLpGroup/>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X79C44528864044C5">2.3-3 GeneratorsOfGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.s class="<> =nocss;<span=chap2#X84E7A9E07A5DFDCF3IsAscendingLPresentationspan
<span class="ContSS"><br /><span class="nocss">  </span><a hrefspanclassContSS/ =""> &;<span<hrefchap2X783B99E381C5C8BF5 EmbeddingOfAscendingSubgroup>
<divd>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X847047F083826C00">2.4 <span class="Heading">Accessing an L-presentation</span></a>
>
<div class="ContSSBlock">
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<span class="ContSS"><br /><spanspan=">br />
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<div>21< =""><span>
<<p>Let <spanclass""S/>beclass"Q/> and besubsetsthe class""F_SSimpleMathΦ set  span>φgtF_Sspan em/ a  span="">S,ΦR</> andit calledemfiniteem>if the setsspan="SimpleMath">S/span,< class="SimpleMath">Q/span>, <span class"SimpleMath>Φ/>, > finiteAfinite L-presentationspanclass"impleMath>(,,,R)definesthe<>finitely) em>L-presented group
<div class="ContSSBlock">
<span class= ="pcenter">=left Q\cup{\varphi\in\Phi}\\right^F_S/>
<span class="ContSS"><br /><span class="nocss">  </span>pis< classSimpleMath>φ</span>-invariant for each <span class="SimpleMath">φ∈</>; that is  < class="">K<> satisfies<spanclassSimpleMath>φ K/> for <span class=SimpleMathφ∈Φ/span> Note   ascending  invariant   eachL-presentation< classSimpleMath(,ΦR<spanthereis   <em  </> span="SimpleMath">S∅ΦR)/span> which is . In itis decidable  ornot  L-presentation isinvariant  thiswouldrequire  solutiontothe .</p>
<span class="ContSS"><br /><spanp>In the  of manual  L-presentedgroup alwaysfinitelyjava.lang.StringIndexOutOfBoundsException: Range [87, 77) out of bounds for length 92
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X87F0C52978D99BB5">2.5-4 IsInvariantLPresentation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X783B99E381C5C8BF">2.5-5 EmbeddingOfAscendingSubgroup</a></span>
</div></div>
< =ContSect ="" /span<">6< =Headingfor <span
</span>
<div class="ContSSBlock">
<span classp>returns < classpkg<>L-presented  the  group<=Arg, relators=Arg></, theo <varArg/, therelatorsvar""<varThe variables <<class"> and< class""irels/> need to of the underlyingfreegroup">endos/> needsto beafinitelist homomorphismsspan="SimpleMath>&;<><p <span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7972B0D87EF36536">2.6-2 SplitExtensionByAutomorphismsLpGroup</a></span> <span class="ContSS"><br /><span class="nocss" <span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X856F237B7BAC3BC8">2.6-4 IsomorphismLpGroup</a></span> </div</div> </div> <h3>2 <span class="Heading">An Introduction to L-presented groups</span></h3> <p><a id="X84541F61810C741D" name="X84541F61810C741D"></a></p> <h4>2.1 <span class="Heading">Definitions</span></h4> pLet<spanclass""><span  analphabet spanclass"">Q</>  spanclass=SimpleMath>R</>besubsets of the  < =SimpleMath>F_S</span  thisalphabet,and class""></span   offree endomorphismsspan="SimpleMath>: &; emL-presentation< < ""(Q,Rspan> andit iscalled<>"<>, span ="""<> finite.A)S,,R)/span> definesthe <>finitely em> group/> <p class="pcenter"> G=\left\langle S \left|  Q\cup \bigcup_{\varphi\in\Phi^*}R^\varphi\right.\right\rangle</p> <p>where <span class="SimpleMath">Φ^*</span> denotes the free monoid generated by <span class="SimpleMath">Φ</span>; that is, the closure of <span class="SimpleMath">Φ∪{ id}</span> under composition.</p> <p>The elements in <span class="SimpleMath">Q</span> are the <em>fixed relators</span class="GAPprompt">gap></span> <span class="GAPinput">irels:=[Comm( d, d^a ,ac***a)];<> <p class="pcenter">K=\left\langle Q\cup \bigcup_{\varphi\in\java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 <p>is <span class="SimpleMath">φ</span>-invariant for each <span class="SimpleMath">φ∈Φ</span>; that is, if <span class="SimpleMath">K</span> satisfies <span class="SimpleMath">K^φ⊂ K</span> for each <span class="SimpleMath">φ∈Φ</span>. Note that every ascending L-presentation is invariant and for each L-presentation <span class="SimpleMath">(S,Q,Φ,R)</span> there is a unique <em>underlyingpreturnssome well-knownexamples finitely groups. The of function  be  positive   <panclass=SimpleMath>0>.< <p>In the remainder of this manualdd <p><a id="X81065E797A486D0F" name="X81065E797A486D0F"></a></p> <h4>2.2 <span class="Heading">Creating an L-presented group</span></h4> <p>The construction of an L-presented group is similar to the construction of a finitely presented group (see Chapter <a href="../../../ <p><a id="X7BBBE4C082AE4D5A" name="X7BBBE4C082AE4D5A"></a></p> <h5>2.2-1 LPresentedGroup</h5> <div class="func"><table class="func" width="100%"><tr>dt< class"">n=5</></dt <ing free group <var class="Arg">F</var,the relators< class="Arg">frels/var>, the set endomorphisms<var="">ndosvar,and iterated < classArg</varThevariables class><var <var="irels/> to be finite subsets the var="">F andArg</> needs be list  <span=SimpleMathF-gtF/>.<p <p>For example, the Grigorchuk group,</p> <p class="pcenter"> \Big\langle a,b,c,d \Big| a^2,b^2,c^2,d^2,bcd,[d,d^a]^{\sigma^n},[d,d^{acaca}]^{\sigma^n},(n\inℕ_0) \Big\rangle,</p> <p>can beconstructedas./pjava.lang.StringIndexOutOfBoundsException: Range [37, 38) out of bounds for length 37 > <div class=<dd<;.< =".htmlbiBSidki87>Sid87]]and >[BEH08]</a>,</p>
<span class="GAPprompt">gap></span> <span class="GAPinput">F:=FreeGroup( "a", "b", "c", "d" );</span>
<free group on the/dd
<span>< ="Markn8
#I  Assigned the global variables [ <dd<>anindex- class=SimpleMath>3<span subgroup of the Gupta-Sidki group,</p>
<span class="GAPprompt">gap></span> <span class="GAPinput">frels:=[a^2, b^2, c^2, d^2, b*c*d];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput"dt<strong=Mark"n=/strong>/dt>
<span class="GAPprompt">gap></span> <span class="GAPinput">irels:=[Comm( d, d^a ), Comm( d, d^(a*c*a*c*a) )];;</span>
< class>&;<span classGAPinputG:=LPresentedGroup  frels , irels<span
<L-presented group on the generators [ a, b, c, d ]>
</pre></div>

<p>There are various examples of finitely L-presented groups availabledd

<><="X79A034B8851444C9 ="">

<h5>2.2-2 ExamplesOfLPresentations</h5>

<div class="func"><table class="func" width
<preturnssomewell-known examplesof L-presented. Theinput ofthis needsto a positive  most< class"">10/>.</p>

<dl>
<dt><strong class="Mark"n=1</trong/dt>
<>pThe  on generators;cf.java.lang.StringIndexOutOfBoundsException: Range [51, 50) out of bounds for length 221

</dd>
<dt><strong class="Mark">n=2</strong></dt>
<dd><p>the Grigorchuk group on 3 generators; cf. <a href="chapBib.html#biBGrigorchuk80">[Gri80]</a>, <a href="chapBib.html#biBLysenok85">[Lys85]</a>, and <a href="chapBib.html#biBBartholdi03">[Bar03, Theorem 4.6]</a>,</p>

</dd>
<dt><strong class="Mark">n=3</strong></dth5.4FreeBurnsideGrouph5>
<><p>the  < class>/2≀/span. <hrefhtml"Bar03 Theorem .]/java.lang.StringIndexOutOfBoundsException: Index 139 out of bounds for length 139

</dd>
<dt><strong class="Mark">n=4</strong></dt>
<dd><p>the Brunner-Sidki-Vieira group; cf. <a href="chapBib.html#biBBrunnerVieiraSidki99">[BSV99]</a> and <a href="chapBib.html#biBBartholdi03">[Bar03, Theorem 4.4]</a>,</p>

</dd>
<dt><strong class="Mark">n=5</strong></dt>
<dd>pthe supergroup cf href="chapBib.html#biBBartholdiGrigorchuk02">[BG02a  <a href"hapBib.#>Bar03, Theorem4.6

</dd>
<dt><strong class="Mark">n=6</strong></dt>
<dd><p>the Fabrykowski-Gupta group; cf. <a href="chapBib.html#biBFabrykowskiGupta85">[FG85]</a> and <a href="chapBib.html#biBBEH08">[BEH08]</a>,</p>

</dd>
<dt><strong="Mark">n7/></dt>
<dd><p>the java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 0

java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 5
<>strongclass"Mark">n=8<strong</>
<dd><p>an index-<span class="SimpleMath">3</span> subgroup of the Gupta-Sidki group,</p>

</dd>
<dt><strong class="Mark">n=9</strong></dt>
<dd><p>the Basilica group; cf. <a href="chapBib.html#biBGrigorchukZuk02">[GtZ02]</a> and <a href="chapBib.html#biBBartholdiVirag05">[BV05]

</dd>
<dt><strong class="Mark">n=10</strong></dt>
<d>pBaumslag finitely,infinitely  with ;.< =".#biBBaumslag71"[au71><p>

</dd>
</dl>
<p>Furthermore,every free inavariety groups satisfying finitelymany identitiesis  L-presented. of groups  availablefrom thestrong""lprespackagethe;for   referthe  <hrefchapBibhtml#biBH08>Har08</a>.</p>

<p><a id="X7DA323A87E7B6A7C" name="X7DA323A87E7B6A7C"></a></p>

<h5>2.2-3 FreeEngelGroup</h5p>returnsfiniteL-presentation forthe lamplighter  on < class="Arg">int</var lamp states in firstcaseif <var="Arg"filtervar  the filter <code="code">IsLpGroup<code.Inthe ,thegroup<var""pcgroupvar be finitecyclic.Then      for group <odeclass">Sizepcgroup) lamp; for details on the L-presentation see ahref"chapBib.htmlbiBBartholdi03">[Bar03] <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FreeEngelGroupltL-presented grouponthe generators [ a,t,  ]> <p>returns an L-presentation for the free <var class="Arg">n</var>-Engel group on <var class="Arg">num</var> generators; that is, the free group in the variety <L-presented group thegenerators[  a,  t  ]gt <p><a id="X81C3537083E40A5C" name="X81C3537083E40A5C"></a></p> <h5>2.2-4 FreeBurnsideGroup</h5> < class="func"><table class"func width=10%><>td =tdleft>code ="unc"̻ FreeBurnsideGroup < =Arg"num/ar,<var class"">exp/var> )</><td ="tdright"( operationnbsp)/></tr<table<div
<p>eturnsan for free  groupon<ar class="">numvargeneratorswithexponentvar=Arg">exp; that is, free group on num generators inon ><var</> <p><a id="X8796306C7A7924D1" name="X8796306C7A7924D1"></a></p> <h5>2.2-5 FreeNilpotentGroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 <p>returns an L-presentation for the free nilpotent group of class <var class="ltinvariant on the [ C12) C(,3), C(21, C23, (,) (,), M1[,], <p><a id="X81450ABA81F0FCE5" name="X81450ABA81F0FCE5"></a></p> <h5>2.2  M2[,], (3[,] ]> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneralizedFabrykowskiGuptaLpGroup class"gapgt;span < class">ank3> <> an for <var="Arg>-hgeneralizedFabrykowski-Gupta groupconstructed a=chapBib.#iBBEH08"[<a<pjava.lang.StringIndexOutOfBoundsException: Index 167 out of bounds for length 167 <p><a id="X83BF8C597E1DC266" name="X83BF8C597E1DC266"></a></p> <h5>2.2-7 LamplighterGroup=&<span="GAPinput">r =AbelianInvariants[i]lcs+1)<span < class< class"" width10"><>

tdleft="func"̻ </code> var="Arg>ilter/>, (&;operation )/>

width10%"<>< class=tdleft>code class"func&827;FreeGroupOfLpGroupcode( <var="Arg">lpgroupvar)/tdtd="tdright">&;attributenbsp;)/></tr</table/>
&;groupofsize withgenerators&;
<span class="GAPprompt">gap></span> <span class="GAPinput">ia := Source(EmbeddingOfIASubgroup(a));</span>
<invariant LpGroup on the generators[C(12), C(1,), (,1), C(2,3) C3,), C(32, M(1,[,3]),
  M(2,[1,3]), M(3,[1,2]) ]>
< class="GAPprompt">gap><span<pan="GAPinput">rank: 3</pan
3
< =""gap/>< =GAPinput =NilpotentQuotient,rank/pan
<span class="GAPprompt">gap> free which theL-presented < class"Arg">lpgroup><p
<span class
span">gt;span spanclass""> r: AbelianInvariants([i]/lcs[i+1);/>
<span class="GAPprompt"
<span="APprompt>gt<> Print"quotient invariants collected,\)/>
<span class="GAPprompt">>
st    ,  ,,,   ,9]]java.lang.StringIndexOutOfBoundsException: Index 87 out of bounds for length 87
2 quotient  [ 0 0,0 0, 0,0, ,0 , 0 0,0 , , 0 ,0 0
] (collected [ [ 0, 18 ] ])
3rd quotient: abelian invariants [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
  ,0 ,0 ,0 0 0,,0 ,0 ,0 00 0 0 ,0 ,0, ,2 2 2, 2, 2 2 22 2
  2, 2, 2, 2, 3, 3java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</pre></div>

<p><a id="X80B65AF48662DE70" name="X80B65AF48662DE70"></a></p>

<h4>2.3 <span class="Heading">The underlying free group<

<p>An L-presented group is defined

<p><a id="X7F883CC57A3CCAC7" name="X7F883CC57A3CCAC7"></a></p>

<h5>2.3-1 FreeGroupOfLpGroup</h5>eturns element theL-presented represented the < class"elm/ar on theof underlyingfreegroup,if">fam the family of L-presented group elements./>

<divdiv=example>
p: underlying java.lang.StringIndexOutOfBoundsException: Range [40, 37) out of bounds for length 97

<p><a id="X838079A587E8CF43" name="X838079A587E8CF43"></a></p>

<h5>.3-2FreeGeneratorsOfLpGroup>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FreeGeneratorsOfLpGroup</code>( <var class="Arg">lpgroup</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: thejava.lang.StringIndexOutOfBoundsException: Range [0, 16) out of bounds for length 10

<p><a id="X79C44528864044C5" name="span class="GAPprompt">gap> ">last =last2;

<h533GeneratorsOfGroup/5java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32

<div=functable =00"<codeclass""&827>pgroup> )/d>tdclass">&;attribute

/able

<p>Returns: the generators of the L-presented group <java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 4



<p><a id="X85C405D57F65048A" name="X85C405D57F65048A"></a></p>



<h5>2.3-4 UnderlyingElement</h5



<div class="func"><table java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p>returns the preimage of an L-presented group element <var class="Arg">elm</var> in the underlyingjava.lang.StringIndexOutOfBoundsException: Index 101 out of bounds for length 0



<p><a id="X8573CDF57CB216D7" name="X8573CDF57CB216D7"></a></p>



<h5>2.3-5 ElementOfLpGroup</h5>



<div

<> the the represented wordclass></> on generators the group< classfam>is family group.<p





<div="">>

<span class="GAPprompt"

span=GAPprompt>/> <span classGAPinput:LPresentedGroup(   .^ ]  ( F , . ];<span

<spanjava.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 0

true

<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfGroup( G );</span>

[ f1, f2 ]

<span class="GAPprompt">gap></span> <spanpReturns the iteratedrelators ofthe L-presentedgroup <var class"">lpgroup/> as elements the underlying group.</pjava.lang.StringIndexOutOfBoundsException: Index 135 out of bounds for length 135

[ f1, f2 ]

<span class="GAPprompt">gap></span> <span class="GAPinput">last = last2

false

<span class="GAPprompt">gap></span> <span class="GAPinput">UnderlyingElement( G.1 );</span>

f1

<span class="GAPprompt">gap></span> <span class="GAPinput">last in F;</span>

true

<span class="GAPprompt">gap></span> <span class="GAPinput">ElementOfLpGroup( ElementsFamily( FamilyObj( G ) ), last2<iv class"example"<pre>

java.lang.StringIndexOutOfBoundsException: Range [45, 4) out of bounds for length 4

<pre/iv



<="X847047F083826C00"nameX847047F083826C00<a<p>



<h4. span="Heading">Accessing  L-presentationspan/>



<[f1



<p>aid"="X7CD9BE57815552FF>/



<h5>2.4-1 FixedRelatorsOfLpGroup</h5=""gapspan span="GAPinput">EndomorphismsOfLpGroupG)<span



div="func">tableclasswidth"0"<><td="tdleft">code="func">&/code>( var="Arg">lpgroupvar/><td="tdright">(nbsp >/<></>

<p>Returns: the fixed relators of the L-presented group <var



<p><a id="X7C468D1C81964268" name="java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0



<h5>2.4-2 IteratedRelatorsOfLpGroup</h5>



<div class="func"><table class="func<>For the method-selection  of the nilpotent quotient algorithm anL-presented group  have the following attributes and properties.

<p>Returns: the iterated relators of the L-presented<p>< id"X85E77B29796AB730" name="X85E77B29796AB730"></a></p>



<p><a id="X85D253888263A3F6" name="X85D253888263A3F6"></a></p>



<h5>2.4-3 EndomorphismsOfLpGroup</h5>



<div class="func"<tableclassfuncwidth10%><trtd class=tdleftcode classfunc&82; EndomorphismsOfLpGroupcode>( var="Arg">lpgroup/> )</tdtd="tdright">&;attribute&;)</></tr><table></div>

<p>Returns: the endomorphisms





<div class="example"><pre>

<span="GAPprompt">ap;</pan classGAPinputF=( 2 ;<span

<span class<p>< id="" name"X86F017E085082624">/>/>

<L-presented group on the generators [ f1,<>2.52 UnderlyingInvariantLPresentation>

<span class="GAPprompt">gap></span> <span class="func"< class="func width10%>tr=tdleft =func82;UnderlyingInvariantLPresentation< class">lpgroup< class="dright;attribute;/></><></>

[ f1^2 ]

<span class"APprompt"gap<span<span""IteratedRelatorsOfLpGroup<>

[ f2 ]

span""&;/>  classEndomorphismsOfLpGroupG ;span

[ IdentityMapping   be  < class</> For instancethe group

</pre></div



<p><a id"" nameX817DA8E686311B54"



<h4>2



-  may   and<>



<p><a id="X85E77B29796AB730" name="X85E77B29796AB730"></a></p>



<>2.- </h5



< =func=func"0"<>td"tdleft"><codeclassfunc>̻ UnderlyingAscendingLPresentation(<var""lpgroup =tdright(nbsp&bsp>/r<tablediv

<p>returns the underlying ascending L-presentation of <varspan="GAPprompt">></span> <span="GAPinput">AssignGeneratorVariablesF)</span



p< id"="X86F017E085082624a<p>



<>25-2UnderlyingInvariantLPresentationh5>



<div class="func><var is if <var="Arg"is L-presented by class"">S,Φspanthen method to find a subset class">' Q such  spanclass=SimpleMath">(,',ΦR)/pan> is invariantL-presentation.Note there  the ascending <span="SimpleMath">S,ΦR)/>. However, for efficiency the quotient it important the <span="SimpleMath>Q'/> is as big  possible./p>



<p>Since it is undecidable, in general, whether or not a given L-presentation is invariant, there is no algorithm which can determine the best possible underlying invariant L-presentation. The method implemented for this attribute tries to compute a ``good'' invariant L-presentation and will return the underlying ascending L-presentation in the worst case.</p>



p manually>/. ,  p



<p class="pcenter"> \Big

   [,{acacasigmanℕBig\,</pjava.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 54



<p>is invariantly L-presented and ="func"<table="func" ="10%"><tr><td class="tdleft"><code class="func">‣ IsAscendingLPresentation</code>( <var class="Arg">lpgroup</var> )</td><td class="tdright">( property )</td></tr></table></div>





<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">F:=FreeGroup( "a", "b", "c", "d" );;</span>

<span class="GAPprompt">gap></span> <span class="GAPinput">AssignGeneratorVariables( F );</span>

#I  Assigned the global variables [ a, b, c, d ]

<span class="GAPprompt">gap></span> <span class="GAPinput">frels:=[ a^2, b^2, c^2, d^2, b*c*d ];;</span>

<span class="GAPprompt">gap></span> <span class="GAPinput">endos:=[ GroupHomomorphismByImagesNC( F, F, [ a, b, c, d ], [ c^a, d, b, c ]) ];;</span>

<span class="GAPprompt">gap></span> <span class="GAPinput">irels:=[ Comm( d, d^a ), Comm( d, d^(a*c*a*c*a) ) ];;</span>

<span class="GAPprompt">gap></span> <span class="GAPinput">G:=LPresentedGroup( F, frels, endos, irels );</span>

<L-presented group on the generators [ a, b, c, d ]>

<span class="GAPpromptchecks whether the L-presentation of <Arglpgroup/varis ascendingthat if the  is This  whenan groupnorelatorsthe< ""LPresentedGroup>< href.html#X7BBBE4C082AE4D5A> =R">2.2-

/><d>



<p><a id="X84E7A9E07A5DFDCF" name="X84E7A9E07A5DFDCF"></a></p>



<h5>2java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0



div=func<table="func" width="10%"><tr>td class="dleft">codeclassfunc>#827;IsAscendingLPresentationcode(<var="rg">lpgroup</var> )</d><d class="tdright">&; /></tr</table</>

<p>checks whether the L-presentation of <var class="Arg">lpgroup</var> is ascending; that is,java.lang.StringIndexOutOfBoundsException: Range [0, 94) out of bounds for length 0



<p><a id="X87F0C52978D99BB5" name="X87F0C52978D99BB5"></a></p>



<h5>2.5-4 IsInvariantLPresentation</h5>



<div class="

<p>attempts to check whether the L-presentation of <var class="Arg">lpgroup<4>26 <panclass"Heading"Methodsfor L-presented groups/span></h4>



<p><a id="X783B99E381C5C8BF" name> test  L-presented  implemented the<="func">NqEpimorphismNilpotentQuotient><a="..../.pkgn.5.1//chap3.html#"><span="RefLink>nq NqEpimorphismNilpotentQuotient/span>/a>) to comparethe images in  nilpotent   the group. The implemented method successively increases the class of the considered quotient until the images differ. Hence, this method may not terminate and it will only determine whether the elements are different.



<h5>2.5-5 <p><a id="X7C81CB1C7F0D7A90=""></a></



<div class=h5-EpimorphismFromFpGrouph5

<p>stores an embedding of class"unc"table="" ="0%>tr< class=tdleft>codeclass="">

varclass"rg</>, var="rg>n">(&;operation )

<p>aidX7B5C48EA7CD8A57E="X7B5C48EA7CD8A57E">/>/>

<h4>2.6 <span class="Heading">Methods for L-presented groups</span></h4>

<p>Some operations are natural extensions of the operations for finitely generated groups. For example, <code class="code">MappedWord(x,gens,imgs)</java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p>Equality test of elements of L-presented groups is implemented using the operation <code class="func">NqEpimorphismNilpotentQuotient</code> (<> anL-presentation the extensionof<ar="Arg">lpgroup</var anL-presented or afinitely group classArg<>. The of ofvar="Arg>auts.Thus for eachgenerator ></a><p> <h5>2.6-1 EpimorphismFromFpGroup</h5> <div="">table="" width"10">tr class"unc87 /odevarArg<> < =Arg<var/>td"dright">(nbsp;operationnbsp/d>/r<table> <p>returns an epimorphism from a finitely presented group onto <var class="Arg">lpgroup</var>. The finitely presented group is obtained from <var class="Arg <p><a id="X7972B0D87EF36536" name="X7972B0D87EF36536"></a></p> h52</h5> <div class="func"><table class= classgap<> class>( U, H, [aut</span <p>returns an&tL-presented on the generators ,u v, a ]gt <div class="example"><pre> <>a=X84F112247DA4037C"4F112247DA4037C<a>/p> <free group on the generators [ a ]> <span class="GAPprompt">java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 fp group on the generators [ a &; <span class="GAPprompt">gap></span> <span class="GAPinput">U := ExamplesOfLPresentations( 8 );p>returnsanascending L-presentationfor finitely presented group varclass"></ar>orfor a groupvar =Arg"<var./p> <L-presented group on the generators [ t, u, v ]> <span class="GAPprompt">gap></span> <span class="GAPinput">aut:=GroupHomomorphismByImagesNC(<p>< id"X856F237B7BAC3BC8 name="86F237B7BAC3BC8>/a><pjava.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62 [ t, u, v ] -> [ u, v, java.lang.StringIndexOutOfBoundsException: Index 128 out of bounds for length 123 <L-presented group on the generators [ t, u, v, a ]> <pre<div <p><a id="X84F112247DA4037C" name="X84F112247DA4037C"></a></p> <h5>2.6-3 AsLpGroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AsLpGroup</codeltfreegroup generators ] <p>returns an ascending L-presentation for a finitely presented group <var class="Arg">G</var> <p><a id="X856F237B7BAC3BC8" name="X856F237B7BAC3BC8"></a></p> <h5>2.6-4 IsomorphismLpGroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code <span class="GAPprompt">gap></span> <span class="GAPinput">Display(last);</span <p>returns an isomorphism from a finitely presented group <var class=fixed relators=[] <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">F:=FreeGroup( 2 );</span> <free group on the generators [ f1, f2 ]> <span class="GAPprompt">gap></span> <span class="GAPinput">-1*^1f1f2] <fp group on the generators [ f1, f2 ]> <span class="GAPprompt">gap></span> <span class="GAPinput">IsomorphismLpGroup( G );</span> [ f1, f2 ] -> [ f1, f2 ] <span class="GAPprompt">gap></span> <span class="GAPinput">Range(last);</span> <L-presented group on the generators [ f1, f2 ]> <span class="GAPprompt">gap></span> <span class="GAPinput">Display(last);</span> generators = [ f1, f2 ] fixed relators == [ endomorphism = [ IdentityMapping( <free group on the generators [ f1, f2 ]> ) ] iterated relators = [ f1^2, f2^2, f1^-1*f2^-1*f1*f2 ] </pre></div> <div class="chlinkprevnextbot"> <a href="chap0.html">[Top of Book]</a> <a href="chap0.html#contents">[Contents]</a> <a href="chap1.html">[Previous Chapter]</a> <a href="chap3.html">[Next Chapter]</a> </div> <div class="chlinkbot"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a> <a href="chap1.html">1</a> <a href="chap2.html">2</a> <a href="chap3.html">3</a> <a href="chap4.html">4</a> <a href="chap5.html">5</a> <a href="chap6.html">6</a> <a href="chapBib.html">Bib</a> <a href="chapInd.html">Ind</a> </div> <hr /> <p class="foot">generated by <a href="http://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc">GAPDoc2HTML</a></p> </body> </html> quality100% ¤ Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.0.37Bemerkung: ¤ *Bot Zugriff

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