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<?xml version="1.0" encoding="UTF-8" ?>DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd " >html xmlns="http://www.w3.org/1999/xhtml " xml:lang="en" >head >title >GAP (FR) - Chapter 2: FR package</title >meta http-equiv=content-type content"text/html; charset=UTF-8" />"generator" content"GAPDoc2HTML /> "text/ss href=manualcss" />script <tml xmlns"http://www.w3.org/1999/xhtml " xml:lang="en" >script type"text/javascript" >overwriteStyle)<script meta ="generator" contentGAPDoc2HTML/>" div class="chlinktop" ><span class="chlink1" >Goto < src="manual.js" type"text/javascript" ><script div /head >"chlinktop" >span ="chlink1> Chapter: a " .html</><ahref
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href.html ">3 >6</a> <a href="chap7.html" >7</a> <a href="chap8.html" >8</a> <a href="chap9.html" >9</a> <a href="chap10.html" >10</a> <a href="chap11.html" >11</a> <a href="chapBib.html" >Bib</a> <a href="chapInd.html" >Ind</a> </div >div class="ChapSects" ><a href="chap2.html#X7ADCE68284FB4ACF" >2 <span class="Heading" >FR package</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2.html#X80C332C686212786" >2.1 <span class="Heading" >A brief mathematical introduction</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2.html#X78DF4DE18260BD80" >2.2 <span class="Heading" >An example session</span ></a>span >div >div >span class="Heading" >FR package</span ></h3>"X80C332C686212786" name="X80C332C686212786" ></a></p>span class="Heading" >A brief mathematical introduction</span ></h4>, and wish to see their usage within <strong class="pkg" >GAP</strong > through a sample session, please skipskipto <a href="chap2.html#X78DF4DE18260BD80" ><span ="RefLink" >.2</></a>. For more introductionon self-similargroups <a href=chapBibhtml biBMR2091700>[BGN03<a orahrefchapBibhtml #biBMR2035113>BGŠp id="mathjaxlink" class="pcenter" ><a href="chap2_mj.html" >[MathJax on]</a></p>span class="SimpleMath" >G</span > defined by their action on a regular rooted<>< idX7ADCE68284FB4ACF="X7ADCE68284FB4ACF>/>/> "" ><span the < ="" X<>. span "" a∈/>, may to="" X⊂X^<, obtain a permutation <span class="SimpleMath" >π_a</span "X, called ∈<> tree classSimpleMath ∈<span the> of ="" </> span ">x/>, bythe formula/> "pcenter" >(v<spanjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7span span ="" mathematical/><h4span class="SimpleMath" >a_x</java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0span class="SimpleMath" >a ∈ W</span > may java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0em >Moore machinespan ="" ><span is permutational product class>^⋊Sym)/>./"" ></> as restriction the ofspan =<> the< =SimpleMath*span < =SimpleMath>is on of form =SimpleMath^sby state permuting subtrees < ="" _aspan wreathiteratedstates=SimpleMath/panto span ">a_v/span> X^/span./> "≤<> level-transitive if its action is transitive onSimpleMath>^< v ^ isanon-trivial < =" "∈<> that fixes< class" ">X*∖ vX^branched∈Nspan group generated all < =" "for < =" ">has finiteindexin"G/pan.p span vertex the , class/> ':Q×&;Qspan < =SimpleMath>(x/> endpoint at< =SimpleMath>span <=""<;anda span="π of bythe symmetricgroup< =><.We Mealy/> ``' <classϕ&;X<,both class<span ="< /> OfBoundsException: Index 0 out of bounds for length 0span class="SimpleMath" >a</span > were invertible"X78DF4DE18260BD80" name="X78DF4DE18260BD80" ></a></p>span class="Heading" >An example session</> following propertieshave been span ="G≤WspanisG -subsets Xn/>.It is<> branched is < =" >∈> < classSimpleMath>^ ^*/pan <></em > if furthermore for each <span > generated span a_v>for ="<> length spanclass=SimpleMathn/> hasfinite index in < class=SimpleMath div class="example" ><pre >gth 87pre d>sException: Index 12 out of bounds for length 12span ="GAPprompt" >><span span ="GAPinput" >B := FRGroupa=ltb&t(,),b=ltgt);</span span ="GAPprompt" >gap;</span < classGAPinput(B);/>pre ></div >"" B</> We compute of:<pspan classor length 53pre ></div >x 53 out of bounds for length 53code class="func" >IsFRElement</code > (<a href="chap10java.lang.StringIndexOutOfBoundsException: Range [79, 80) out of bounds for length 0 span class="SimpleMath" >{1,2}</span >. The action is computed in the standard manner:</p>div class ="" >pre span =">^ 1^;[,]a pre span =GAPinput (1],2)</>div class="example" ><pre >span class="GAPprompt" >gap></span > <span class="GAPinput" >v := PeriodicList([1],[2[/2 java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6span computationsare more if classSimpleMathB<span >s elements converted <em Mealy<e><p>span class="GAPprompt" >gap&span ="GAPprompt" >ap>/pan class"APinput>m: Image(IsomorphismMealyGroup(B)); length 32div div class="example" ><pre >span >div "GAPprompt" >>>< classa=.1 : .2;s>pre ></div >ngth 4div class="example" ><span "" >>class();/>span class="GAPprompt" >gap></span > <span class="GAPinput" >Order(<pan=GAPprompt&;<span class(;;</>span span ="SimpleMath" Bspan branchedmoreprecisely, the subgroup classSimpleMath<> span "'elementsareconstructed class" "VertexElement/>( =" .#" class" ">415/pan>/>:/> span class="GAPprompt" >gap class"GAPprompt" >gapgtspan classGAPinput =Comm,)<span pre >span class [ 1 .. 2 ] with 3>div class="example" ><pre >span class="GAPprompt" >gap></span > <span classspan class="GAPprompt" >gap></span > <span class="GAPinput" >K := NormalClosure(Bm,Group(c));</span span < class="APinput>(,)Comm(,^2;/> span class="GAPprompt" >gap></span > <span class="GAPinput" >VertexElement(1,c) in K; VertexElement(1,c) in K;</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >DecompositionOfFRElement(VertexElement(1,c))=[[c,One(Bm)],[1,2]];</span >span class="java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 pre ></div >span ="SimpleMath" >[b^,(bp)ap]=<span <span classSimpleMath[a^p,a2)b^</span > for<span class="SimpleMath" >p</panany of <span ="SimpleMath" >2/span <pjava.lang.StringIndexOutOfBoundsException: Index 228 out of bounds for length 228div class="java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4 span class="GAPpromptSince the group isbounded,it iscontracting.Wecompute its nucleus:
"GAPprompt>gap> NucleusOfFRSemigroup;<span pre ><div div class="example" ><pre >span class="GAPprompt" >gap></span > <span class="GAPinput" >NucleusOfFRSemigroup(B)java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0pre class"" gapspan =GAPinputDrawspan th 0div class"" pre =>(DualMachine)/>span span ="" >gapgt;/> <span ="" >M:= AsMealyMachine(a))[1;/>span class="GAPprompt" >gapspan ="GAPprompt" >><span span ="GAPinput" >Draw(DualMachine)4;/span >pre ></div >">. However we asmaller Mealy machinewithstatesonlyab , whichwhichgive better images: div ="<> span class="GAPprompt" >gap><<span class"GAPprompt" >gap;</span < class="GAPinput" WordGrowth::=1,,,1]draw/span span class=" span ="GAPprompt" gap;</span < class="APinput>Draw(DualMachine()4;/panjava.lang.StringIndexOutOfBoundsException: Index 92 out of bounds for length 92 div 098500813508278156span class8pre ></div pre /iv"B/> acts atagiven.codeclass" ode,/code is natural <span =>/> actingspan ="" 2<span pointspdiv class="example" ><pre >span class="GAPprompt" >gap>span class=">/>span class() ;<> pre ></div >div class="example" ><pre >span class="GAPprompt" >gap></span > <span class="GAPinput" >J := JenningsLieAlgebra></div ">> List.5,i-;DimensionGrading)hom_components));<span s for length 0pre ></div >"4 " in position <span div class="chlinkprevnextbot" >/>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="chap7.html" >7</a> <a href="chap8.html" >8</a> <a href="chap9.html" >9</a> <a href="chap10.html" >10</a> <a href="chap11.html" >11</a> <a href="chapBib.html" >Bib</a> <a href="chapInd.html" >Ind</a> </div >hr />"foot" >generated by <a href="https://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc " >GAPDoc2HTML</a></p>body >html >quality 99%
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