java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
}java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 2
\]>
<>By, spanclass"SimpleMath">\(\calE)/span is generatedby threecommutingcatspan class"SimpleMath(1\)/span>groups <span class="SimpleMath">\((G \Rightarrow R), (G \Rightarrow Q)\)</span> and <span class="SimpleMath">\((G \Rightarrow H)\)</span>, but it is more convenient to think of <span class="SimpleMathjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<ul>
<li><p><em>front</em><span class="SimpleMath">\((\calE)\)</span>, generated by <span class="SimpleMath">\((G \Rightarrow R)\)</span> and <span class="SimpleMath">\((G \Rightarrow Q)\)</span>;</p>
</li>
</ul>
<>ecause tail, head and embeddingmaps commute,it followsthat <em>up</em><span class="SimpleMath">\((\calE)\)</span>, generated by <span class="SimpleMath">\(
<div class="func"><table class="func" width="100< =f>table =" =100">>td =tdleft>code =">#8227; </> varclassA"C</></< =>&bsp; </d><tr>java.lang.StringIndexOutOfBoundsException: Index 218 out of bounds for length 218
< class=func"<table class=func" width="100%"><tr><td class="tdleft"><code class="func">&#yle='color: green'>8227; PreCat3Group</code>( <var class="Arg">args</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="unc"><ableclass=func"width"100%>tr>td class=tdleft"<code class="unc>#8227; IsCat3Groupcode(<var class"rg>C<var>> /td>td ="dright"> property )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; PreCat3GroupByPreCat2Groups</java.lang.StringIndexOutOfBoundsException: Index 132 out of bounds for length 0
<> functions code"ode>Cat3Group</code> and <code class="code">PreCat3Group</code> normally take as arguments a pair of cat<span class="SimpleMath">\(^2\)</span>-groups [<em>front, left</em>], or a trio of cat<span class="SimpleMath">\(^1\)</span>-groups [<em>front-up, front-left = left-up, left-left</em>].</p>
<p>In subsection <code class="func">AllCat2GroupsIterator</code> (<a href="chap8_mj.html#java.lang.StringIndexOutOfBoundsException: Index 103 out of bounds for length 0
<> list code=c>()pairs/> obtained may alltriples< =""\[j,[k\/><java.lang.StringIndexOutOfBoundsException: Range [167, 166) out of bounds for length 600
<pIn chapter and theprevious one we areinterested incat<spanclass="SimpleMath">\(^2\)</span>-groups and cat<span class="SimpleMath">\(^3\)</span>-groups, and it is convenient in this section to give
<p>A <em>cat<span class="SimpleMath">\(^n\)</java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 0
<ul>
<li><p><span class="SimpleMath">\(2^n\)</span> groups <span class="SimpleMath<span><h4
<p class="center">\[
\calC_{Ajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
G_A \ G_A \etminus \i})
\quad\mbox{for all} \quad A \subseteq java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
\]/p>
<\ {A\i})
</<p
<li><p>These cat<span class="SimpleMath">\(^1\)</span>-groups combine (in sets of <span class="java.lang.StringIndexOutOfBoundsException: Index 104 out of bounds for length 38
</li>
</ul>
<p/java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
<p>Secondly, we give the simplest of the three descriptions, again adapted from Ellis-Steiner <ahref="java.lang.StringIndexOutOfBoundsException: Index 105 out of bounds for length 0
<p>A <em>cat<span class="SimpleMath">\(^n\)</span>-group</em>Our description a<=S>(n)/span>-group as java.lang.StringIndexOutOfBoundsException: Range [93, 92) out of bounds for length 200
<ul>
<li><p><span
</li>
<java.lang.StringIndexOutOfBoundsException: Range [18, 3) out of bounds for length 229
</li>
<li><p>cat<span class="SimpleMath"
<p java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
(circ)\mboxand~ (hcirce
~\mbox{are the identity mapping on}~ \calB, \qquad
[\ker t, \java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 0
\]<( \circ e) ~mbox{nd~ (h \irc e)
<span class="GAPprompt">gap></span> <span class="GAPinput">PC4 :=
(re)cat4-roup withgenerating-)at1-roups: 1 : [d12 => Group( [ (), (1,4)(2,3)(5,6) ] )] 2 :[12=&t;Group( [(1,4)(2,5)3,),(26)3 ] 3 : [d12 => Group( [ (1,5,3)(2,6,4), (1,4)(2,3)(5,6) ] )] 4 :[d12 =&t Group( [(,2,,4,,6), (,6)3,5 )
<span class="GAPprompt">gap>2 :[12=;Group( [(1,)(25)36) 26)3,)])java.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56
true
<span class="GAPprompt">4 :[d12 =&t; Group([(12,3,4,,6) (2,6(,) ])java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 54 5
</pre></div>
<p>For a cat<span java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 4
<div class="example">5
<span class=GAPprompt">ap><span> <pan class="GAPinput">G := Group( (1,2), (3,4,5), (6,7,8,9,10), (11,12,13,14,15,16,17),</span>
<span <p> a cat<pan ="SimpleMath">(5\</pan>-group we startwith the cyclic group whose orderisthe product the firstfiveprimes.With thisgroupwe mayformspan =SimpleMath"\32\<span> cat<span class==SimpleMath">\(^1\)</span>-groups and <span class="SimpleMath">\(528\)</span> cat<span class="SimpleMath">\(^2\)</span>-groups.</p>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetName( G, "C2310" );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">all1 := AllCat1Groups( G );;</span>
java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 133
G has 32 cat1-groups
<class"APprompt"gap&t<spanspan class="GAPinput">PC5 := PreCatnGroup( [ all1[2], all1[5], all1[13], all1[25], all1[32] ] );</span>
(pre-)cat5-group with generating (pre-)cat1-groups: 1 : [C2310 => Group( [ (), (), (), (), (1,2) ] )] 2:[2310 =gt; Group ) ) (),(3,,5) (,2)])] 3; Group [(),(, ,7,8 ,) 345,(1, ] 4 : [C2310 =&span=">&t;/><pan =G>( [[2,all1[] [13,all1[] []])</>
(1,2) ] )] 5:[2310=gt 20,122,,,,262728,9,0),java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
)3,5,(1,)] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">HigherDimension( PC5 );</span> 6
</pre></div>
<div class="chlinkprevnextbot"> <a href="chap0_mj.html">[Top of Book]</a> <a href="chap0_mj.html#contents">[Contents]</a> <a href="chap8_mj.html">[Previous Chapter]</a> <a href="chap10_mj.html">[Next Chapter]</a>&java.lang.StringIndexOutOfBoundsException: Index 254 out of bounds for length 81
<div(,,1314,,617), ,8 ,) 4)12) )
<hr /=";="( ;/java.lang.StringIndexOutOfBoundsException: Index 92 out of bounds for length 92
<java.lang.StringIndexOutOfBoundsException: Range [0, 2) out of bounds for length 0
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