Let ="SimpleMath> group over this alphabet and< class="SimpleMath">Φ be a of free groupendomorphisms . AnL-presentationis quadruple<spanandspan=""></span are. () < ="impleMath">SQΦspan (em/em <L-presented/em/>
<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</em> and the elements in <span class="SimpleMath">R</span> are the <em>iterated relators</em> of the L-presentation <span class="SimpleMath">(S,Q,Φ,R)</span>. An L-presentation of the form <span class="SimpleMath">(S,∅,Φ,R)</span> is an <em>ascending L-presentation</em> and it is an <em>invariant L-presentation</em> if the normal subgroup</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="../../../doc/ref/chap47.html#X7AA982637E90B35A"><span class="RefLink">Reference: Finitely Presented Groups</span></a> of the <strong class="pkg">GAP</strong> Reference manual for further details).</p>
<h5>2.p> well-knownexamples finitely groups input function be integerat span=SimpleMath1<span/
<div class=><s><dt
<p>returns an L-presentation for the free <var class="Arg">n</var>-Engel group on <var class="Arg">num</vardd>>The Grigorchukgroup 4generators;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>
<div class="func"><table class<d><>the lamplightergroup<span="SimpleMath"ℤℤ</>; cf <a ="chapBib.#biBBartholdi03">,Theorem41<a>,</p>
<p>returns an L-presentation for the free Burnside group on <var class="Arg">num</var> generators with exponent <var class="Arg">exp</var>; that is, the free group on <var class="Arg">num</var> generators in the variety of groups with exponent <var class="Arg">exp</var>.</p>
<div class="func"><table class="func"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>returns an L-presentation dt class=<strong
<div class="func"><table class="func" width="100%"><tr><<d<>'s generated, relatedgroup a trivialmultiplier cf ahref=chapBibhtmlbiBBaumslag71>[]
<div class="func"><table class="func" widthFurthermore every group of satisfying many isfinitely Some thesegroupsare fromthe < class="pkg>lpres-package using following operations;for further detailswerefer to diplomathesis< =".html"[Har08]/
< a L-presentation the groupvarArg<>lamp the , if class></> is class"IsLpGroup/> second case the pcgroup mustbe afinite cyclic group themethodreturnsafiniteL-presentation thelamplighter on ><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">LamplighterGroup( IsLpGroup, 2 );</span>
&; the,u]gt
<span class="GAPprompt">gap></span> <span class="GAPinput">LamplighterGroup( IsLpGroup, CyclicGroup(3) );</span>
< on [,,u]&;
</pre></div>
<p
<h5
<div=func="width"0"tr"2/(varclass"><v> < =Arg<vartdclass>nbsp&;<td>/><>
<p>computes an L-presentation for the IA-automorphism group of a free group. This is > L-presentation the Burnside <ar"Arg>/> generators exponent < class"Argvarthe onclass"num/> generatorsinthevariety withexponent var class=Arg>xp/var>.
<p>The L-presentation is taken from <a href="chapBib.html#biBDayPutman">[DP]</a>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">f := FreeGroup(3);</span>
<free group on the generators [ f1, f2, f3 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">a := AutomorphismGroup(f);</span>
<group of size infinity with 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">ia := Source(EmbeddingOfIASubgroup(a));</span>
&;LpGroupgeneratorsC(,) C13, C(,) (,) C31,C32, (,23)java.lang.StringIndexOutOfBoundsException: Index 101 out of bounds for length 101
(,13) M(3,12)java.lang.StringIndexOutOfBoundsException: Range [27, 26) out of bounds for length 30
<span="GAPprompt>>spanclass=GAPinput"> := ;</span
3
<span class="GAPprompt">gap></span> <span class="GAPinput">q := NilpotentQuotient(ia,rankpreturns L-presentation the class"n[EH08]/>./>
<span class="GAPprompt">gap></span> <span class="GAPinput">lcs
<span class="GAPprompt">gap></span> <span class="GAPinput">for i in [java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span class"GAPprompt">&t;</span> <span classGAPinput : AbelianInvariants(lcs/lcs[i+1;/>
<span class="GAPprompt">><<div="func">table=func="100">trclass">><var class">int)/td>
< class="func>class" ="10>trtdclass""< =func">#2;</>< class</> <>< class>nbsp <td>/><div
<p>Returns: the underlying free group of the L-presented group <var class="Arg">lpgroup</varlt of infinity 3 generatorsgt
<p><a;invarianton (,,C13,C2,,(1 3) (,2],
<h5>2.spangap<> < class :;s>
<div class="gap>">q : (ia,rank);;
the group underlies groupvar=Arg</var/>
<div class="func"><table class="func" width="100%"><tr><td1 quotient: abelianinvariants[ 0, 0,0, 0,0, 0 0,0,0 ]](collected[ [0 9 ] ])
<p>Returns: the generators of the L-presented group <var class="Arg">lpgroup</var>. These are the images of the generators of the underlying free groupnd: abelianinvariants ,0 ,0, ,0 ,0 , 0, ,00,0 ,java.lang.StringIndexOutOfBoundsException: Index 87 out of bounds for length 87
<div class
<p> the inthe group by word<ar="Arg>elm onthe generators oftheunderlying var =Arg"fam>is L-presented<pjava.lang.StringIndexOutOfBoundsException: Index 225 out of bounds for length 225
< class""><pre
<span class<>Returns the freegroup of the L-presented group <var class="Arg">lpgroup</var></p>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=LPresentedGroup
<span>2</h5
true
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfGroup( G );</span>
[ f1, f2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">FreeGeneratorsOfLpGroup( G );</span>
[ f1, f2 ]
<=GAPinput <span
false
<span class="GAPprompt">gap>>2.- GeneratorsOfGroup</5>
f1
<span class=func">< class="func"width=10%>tr>"< =func>#2; GeneratorsOfGroup var =Arg"></var)/d>td ="tdright">nbsp;<td>/>/iv
true
<span class="GAPprompt">gap></span> <span class="GAPinput">ElementOfLpGroup( ElementsFamily
true
</pre></div>
<p><a id="X847047F083826C00" name
<h4>2.4 <span class="Heading">Accessing an L-presentation</span></h4>
<p>The fixed relators, the iterated relators, and the endomorphisms of an L-presented group are accessible with the following methods.</p>
<div class="func"><table<p>eturns element in L-presented group by the <var class="Arg"elm/varon the generators of underlying free, if <var="Arg"></var> the of L-presented elements/>
<p>Returns: thediv classexample<pre
">(&;attribute&bsp;)/>
< classgap/span class( ;/>
#I Assigned the global variables [ a, b, c, d ]
<span class="GAPprompt">gap<p>a ="X86F017E085082624"name">/p
<h5.- </>
class"">&;<span=">:=[Comm(d da) ( d, d^(a*c*aca) ) ];
<span>attempts compute good L-presentation varclassArglpgroup/>; that, class>lpgroup</var> finitelyL-presentedby <spanclass=SimpleMath(,,,R)</>, this seekstofind a <span="SimpleMath">'⊆span that< =(,',</pan an L-presentation that isalways underlying L-presentation classSimpleMath(,,,R<span the of nilpotent algorithm is that subset class"'span> aspossible./pjava.lang.StringIndexOutOfBoundsException: Index 648 out of bounds for length 648
<L-presented group on the generators [ a, b, c, d ]>
<span class="
This attribute can be set using etUnderlyingInvariantLPresentation.For instance theGrigorchukgroup
<divclass> classwidth0tr
checks "> ascending; is,if set offixed relators empty.This property issetautomatically creating an L-presented group with fixed relators using the function codeclass=func> (ahref="chap2#">spanclass"efLink-span/>./p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsInvariantLPresentation</code>( <var class="
<p>attempts to check whether the L-presentation of <var class="Arg">lpgroup</var> is invariant. In general, one cannot decide whether or not a given L-presentation is invariant. There are mainly two methods implemented for this property. The first method seeks< class"">< class width0<>td"> =""&87(var class"rgvar/<dclass=""(nbspproperty;)<tdtr>/table>/iv
<p>Equality of elementsof groupsisimplemented usingthe operation <ode classfunc</code (a href/..//q-21docX8758F663782AE655 class":
" name="X7C81CB1C7F0D7A90//p>
<>2.6- EpimorphismFromFpGroup</h5>
<div="unc"< classfuncwidth10"<>tdclass=""< ="func>‣ pimorphismFromFpGroup(< ="">lpgroupvar< class""n/>)/td classtdrightnbspnbsptd>/table
<p>returns< ="X7B5C48EA7CD8A57E" name><a</java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
<p><a java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<div class="func"><tablejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<returns for split < class>lpgroup/> by L-presented by presented <var="">H<var action agenerator < class""H/>onclass">lpgroup<> is givenbyan in the listvar="Argv. of">H<> theremust be an automorphisminthe list < class"">auts
<> ="X7C81CB1C7F0D7A90"name<
<span
<free group on the generators [ a ]>
<span class="GAPprompt">gapdiv classfunc< classfunc="10%"><><td="tdleft>code ="">EpimorphismFromFpGroup( lpgroup">n )>gap><<
<L-presented group on the generators [
<span<>2.6-2 SplitExtensionByAutomorphismsLpGroup
[ t, u, v ] -> [ u, v, t ] span="GAPprompt">><span<spanclass="GAPinput"SplitExtensionByAutomorphismsLpGroupH, ] );/>
&; group [ t u,v ];
</pre></java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p< id"X84F112247DA4037C" name="8F112247DA4037C"></a</pjava.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsomorphismLpGroup</code>(<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsomorphismLpGroup</code>(
<p>returns an isomorphism from a finitely presented group <var class="Arg">G</var> or from a free group <var class="Arg">G</var> to the L-presented group obtained from the method <code class="func">AsLpGroup</code> (<a /pre>/>
<div class="example"><pre>
<span class="java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
&; group on the [f1, f2 ]gt;
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=F/[ F.1^2, F.2^2, Comm( F.1, F.2 ) ];</span>
<span class="GAPprompt">gap></span> <span class="GAPinputjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
[ f1, f2 ]java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span class="GAPprompt">gap></span> <span class="GAPinput">Range(last);</span>
<L-presented group on the generators [ f1, f2 ]>
>
generators = [ f1, f2 ]
java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
endomorphism = [
IdentityMapping( <free group on the generators [ f1, f2 ]> ) ]
iterated relators = [
f1^2,
f2^2,
f1^-1f2-**
</pre></div>
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