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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9_mj.html#X85A5A6967D942463">9.1-2 Front3DimensionalGroup</a></span>
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<<
</div"java.lang.StringIndexOutOfBoundsException: Range [24, 23) out of bounds for length 25

<h3>9 <span class="Heading">Cat<span class="SimpleMath">\(^3\)</span>-groups and Crossed cubes</span></h3>

<p>The term <em>4</iv/div>

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

<p><a id="X7CC52AF4840F478E" name="X7CC52AF4840F478E"></a></p>

491 span=H>for(-span=>^\<-/<>

<p>We shall java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 0

<p class="center">

\vcenter{\
amp  d]&t+5> a[ ;00>
     java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
     a[ ;0.>
   &;& ;; ar lt+.>
             \ar[dddd] &     \ar[dr] <+0.6ex>  amp  a[];.ex;
             [] lt0.gt
           & \d]l;06x;
& a; [ ;0.gt addddl;java.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 64
       a &;0.>a[]&+0>
       \ar[ul] <\[ l;05xg;\rrrr&;0.>
     a; amp a   arllll&;0.g;
                \ar[     & & & & l]l;06>
                ul lt+.& ar]+.&    \
& & & &\ar[ull;0.> a[ lt+.gt;    \\
& & && a;& \
& M \ar[uuuu] <+0.6ex>
     \ar[rrrr] <+& java.lang.StringIndexOutOfBoundsException: Range [14, 11) out of bounds for length 39
     \[ lt+06xg;
   &;& amp ;La[ lt+.&
             \ar[llll] <+0.6ex>
              +.e&
           && a;a;a; ar +.exgt
&\l]&+.&;
       \java.lang.StringIndexOutOfBoundsException: Range [13, 10) out of bounds for length 37
       \ar[ul] <+0  amp  ar l0.g;
     &mp ;a;a  a[]l;06g;
               \ar[llll] <+0.6       \ar[ul] <+0.5ex> \ar[u&t+.e&t
               \r[l <+0.ex>\ru]&t+.0ex&t  \
}}

\]</p>

<p>By definition, <span class="SimpleMath">\(\calE\)</span> is generated by}

<l
<p definition< =SimpleMath>(\<>is  by  <=">\^\<>java.lang.StringIndexOutOfBoundsException: Range [141, 140) out of bounds for length 425

</li>
<li><p><em>left</em><span class="SimpleMath">\((\calE)\)</span>, generated by <span class="SimpleMath">\((G \Rightarrow Q)\)</span> and <span class="SimpleMath">\((G java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

</li>
</ul>
<p>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p><a id="X7828496F7D72E232"

<>the,    all  java.lang.StringIndexOutOfBoundsException: Range [74, 73) out of bounds for length 697

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Cat3Group</java.lang.StringIndexOutOfBoundsException: Index 115 out of bounds for length 0
<div class="func"><table java.lang.StringIndexOutOfBoundsException: Range [0, 30) out of bounds for length 0
<ivclass"unc"< class"func" width"%">tr<class"tdleft"<class"func"&;IsCat3Group/code(<var ="rg></ar>)/d>td class=tdright"(nbspproperty)/d<</table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; PreCat3GroupByPreCat2Groups</code>( <var <ivclass="=java.lang.StringIndexOutOfBoundsException: Range [37, 36) out of bounds for length 222
<p>The global functions <code class="code">Cat3Group</code> and <code class="code">PreCat3Group</code> normally take as arguments a pair of catdiv=f"< " =100"<<=>f">8227; </>(var =""< )<<classt">;java.lang.StringIndexOutOfBoundsException: Range [188, 187) out of bounds for length 218

<p>In subsection <code class="func">AllCat2GroupsIterator</code> (<a href="chap8_mj.html#X7EFCF9697E845B2C"><span class="RefLink">8.6-4</span></a>) the list of pairs <code class="code">CatnGroupLists(d12).pairs</code> contains the three entries <code class="<pThe global< class=c"Cat3Group <java.lang.StringIndexOutOfBoundsException: Range [70, 69) out of bounds for length 332

<div class="example"><pre>

<span class="GAPprompt">gap></span> <java.lang.StringIndexOutOfBoundsException: Range [0, 44) out of bounds for length 0
<panjava.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 112
<"java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 113
<span class="GAPprompt">gap></span> <span class="GAPinput">C3Ga := Cat3Group( C68, C811 );</span>
cat3-group with generating (pre-)cat1-groups:
1 : [d12 => Group( [ (), (1,6)(2,5)<span class"GAPprompt">gap;</span> < class"APinput">C68 := Cat2Group( alld12[6], alld12[8] );; </span>
2 : [d12 => Group( [ (1,4)(2,5)(3,6), (1,3)(4,6) ] )]
3 :[&;([ 15,),,4,14(23(,)])java.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60
< "">>/>  => =(6,8,11]);</span>
<span class="2 : [d12 => Group(4(,)36,(3(,) ]
true

<>div

<p><a =Gjava.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 124

<h5>9.1-2 Front3DimensionalGroup</h5>

<iv">= =100%>tr<class=>code =>#;</code> var =""C3</>)/>td =tjava.lang.StringIndexOutOfBoundsException: Range [183, 182) out of bounds for length 231
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Left3DimensionalGroup</code>( <var class="Arg">C3</var> )</tdtrue
<div class="func"><table java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 0
<iv "> =" =100%>><class> f"&8227;</code>  =></> )/td> =tjava.lang.StringIndexOutOfBoundsException: Range [183, 182) out of bounds for length 231
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Down3DimensionalGroup</code>( <var class="Arg">C3</var> )</td><td class="tdright">( attribute )<<div class"func">< class""width100"<>< classclass="dleft">code class="func">‣ Left3DimensionalGroup</code>( <var class="Arg">C3</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Back3DimensionalGroup</<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Right3DimensionalGroup</code )td><td class=tdright>&;&bsp;/>/>/>/>
<p>The six faces of a cat<span class="SimpleMath">\(^3\)</span>-group are stored as these attributes.</p>

<div class="example"><pre>

<spandiv =f">table class="func =100"<< =tdleft"><code class="func">&#>8227; Back3DimensionalGroup</code>( <var class="Arg">C3</var> )</td><td class="tdright">( attribute p>   ofacatspanclass""\^\<>areasattributesjava.lang.StringIndexOutOfBoundsException: Index 105 out of bounds for length 105
<class"APprompt"gap&t<> spanclass""Up3DimensionalGroupC3Ga  =;/>
true

</pre></div>

<p><a pre></div

<

<p>Once<Once thelist< class"ode"CatnGroupListsG.<code has beenobtained wemay seek  span class"impleMath>(i,][,]\<span>and span class="SimpleMath">\([k,i]\)</span> or <span class="SimpleMath">\([i,k]\)</span> of pairs in this list and then, for each such triple, construct a cat<span class="SimpleMath"

<p><a id="java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 0

<h5>9.2-1 AllCat3GroupTriples</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; AllCat3GroupTriples</code>( <var class="Arg"><div class""> class="width=100"<r =tdleft< =f>#8227 AllCat3GroupsNumber</> <varclass"rg"G<var><td<td =tdright>(nbspa <td><tr>table<div
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; AllCat3GroupsNumber</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )<  of  by the operation<codeclass=code"AllCat3GroupTriples</code> is saved as <code class="code">CatnGroupLists(G).cat3triples</code>. The length of this list is the number of cat<span class="SimpleMath">\(^3\)</span>-groups on <span class="SimpleMath">\(
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 2
<p[5 7,11 ]

<p>As yet =GAPprompt"gap><span> <span class="GAPinput">Cat3Group( alld12[5], alld12[7], alld12[11] );</span>

<div cat3-group with generating (recat1groups:

<span class="GAPprompt2  [12=gt; Group([(1,)2,5)(,) (6(35  ]
<span class="GAPprompt">gap></span> <span class="GAPinput">CatnGroupNumbers( d12 ).cat3; </span>
94
<span class="GAPprompt">gap></span> <span class="GAPinput">triples[46];</span>
7,11 java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
<spanclass=GAPprompt>></><span class="APinput"java.lang.StringIndexOutOfBoundsException: Range [69, 68) out of bounds for length 102
<span class="GAPprompt">gap></span
groupjava.lang.StringIndexOutOfBoundsException: Range [16, 15) out of bounds for length 45
=&t ([( 14),)56)]]
2 : [d12 =>/pan<>
3 : [d12 =>> this        java.lang.StringIndexOutOfBoundsException: Range [89, 87) out of bounds for length 335

</pre></div>

<p><a id="X7F8538B580847268" name="X7F8538B580847268"></a></p>

<h4>9.3 <span class="Heading">
Definition and constructions for cat<span class="java.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 0
<<>

<p>In this chapter and the java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 0

<p>A <em>cat<java.lang.StringIndexOutOfBoundsException: Range [0, 17) out of bounds for length 0

<ul>
<li><p><span class \o{ \\\})

</li>
<li><p>Group homomorphisms forming<pjava.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6

<p class="center">\[
\calC_{A,i} ~=~ (e_{A,i};\; t_{A,i},\; h_{A,i} \ :\
_A \oG_ \setminus \{\},
\quad\mbox
\]/>

<p>the <em>edges</em> of the cube.</p>

</li>
<li><p>These cat<span class="SimpleMath">\(^1\)</span>-groups combine (in sets of

<li>
</ul>
<p>Note that, since the <span class="SimpleMathjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p>Secondly, we give the simplest of the three descriptions, again adapted from Ellis-Steiner <a href="java.lang.StringIndexOutOfBoundsException: Index 109 out of bounds for length 0

<p>A cat<span class="SimpleMath">\(^n\)</java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

< e_{]i  {[]\setminus\i\} \to G_[n}
t_{[],} {[]i} : G_{[n] \to G_{n \ \{\},
e_{[n],ijava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
\</>

<p>satisfying the following axioms for all <span class="SimpleMath">\(1 \leqslant

<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<li>p>  =>\[]} ~~e_[]i;\ {n,}\ {n,}  \{n}\ {  {\}))/  <commutingem catspan class="impleMath"\^)/-roups so that:<p

</li>
<li><p><span class="SimpleMath">\( (e_1 \circ t_1) \circ (e_2 \circ java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

</li>
<li><p><span java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 0

</li>
</ul>
<p> third definesa catspan class"impleMath"\^\< a "cat<span class="SimpleMath">\(^1\)</span>-group of cat<span class="SimpleMath">\(^{(n-1)}\)</span>-groups".</p>

<p>A <em>cat<java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 0

<ul>
<li><p><span li><p><span class="SimpleMath">\(\calB\)</span> with groups <span class="

</li>
<li><p><span class="SimpleMath

</li>
<li><p>cat<span class="SimpleMath">\(^{(n-1)}\)</span>-morphisms <span class="SimpleMatht \ e)~{and} ( \ e)

<p class="center">\[
(\  \{}( c e
~\mbox{/>
[k ,\h  {1\ }
\]</p>

</li>
</ul>
<p><aidX ="C>/></>

<h5>9.3-1 div class="func="=100"<><class""<">8227 </>( class=" << java.lang.StringIndexOutOfBoundsException: Range [163, 162) out of bounds for length 220

<ivclass"func"< class"" width=100%>trtdclass=code class="unc"&>8227;PreCatnGroup</code>( <var =Arg></var>)/><tdclass="dright">(nbsp;peration&bsp;<td>/tr>/table<div
<divclass""<"" ="100"<>=> ">#8227 </> < =Arg></ar> )>td =tjava.lang.StringIndexOutOfBoundsException: Range [169, 168) out of bounds for length 217
<p>The operation <code class="code">(Pre)CatnGroup</code> expects as input a list of java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

<(-- generating  (pre)-java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51

<span class="GAPprompt">gap:d &t  [14(25)36,(,),5)])java.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56
(pre-)cat4-group with generating (pre-)cat1-groups:
4 : g;  12,4,6) (6(,)])
  d >   14(,),)(,)35  ]
3 : [d12 => Group( [ (1,5,3)(2,6,4), (1,4)(2,3)(5,6) ] )]
4 &t   1,3,456)(,)35  ]
<span class="GAPprompt">gap></span> <span class="java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 1
true
<span class="GAPprompt">gap></span> <span class=
5

</pre></span=>;/ <java.lang.StringIndexOutOfBoundsException: Range [51, 50) out of bounds for length 134

For <class>^\/- may the       of      we <class=>(\< "java.lang.StringIndexOutOfBoundsException: Range [239, 238) out of bounds for length 358

<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">G := Group( (1,2), (3,4,5), (6,7,8,9,10), (11,12,13,14,15,16,17),</span>
<span class="GAPprompt<span class="GAPprompt">gap></span> <span class="GAPinput">Print( "G has ", CatnGroupNumbers( G ).cat1, " cat1-groups\n" );</span>
<span classspan ="APprompt"&;/><java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 143
<span class="GAPprompt">gap></span> <span class="GAPinput">all1 := AllCat1Groups( java.lang.StringIndexOutOfBoundsException: Range [0, 86) out of bounds for length 51
<span class="  C2310= ([(,(, ),345)(,2  ]
G has 32  : [C2310 =>([ ) (6 7,8,910, (,,) (1,)])java.lang.StringIndexOutOfBoundsException: Index 71 out of bounds for length 71
< class"GAPprompt">gapgt<span><panclass"APinput"PC5 := PreCatnGroup [all12] 5,all113] all125,all132 ];/pan
(pre-)cat5-group with generating (pre :C2310 >Group([(20,,22232425,26,,28,9,0),
1 : [C2310 => Group( [ (), (), (), (), (1,2) ] )]
2 : [C2310 => Group( [ (), (), (), (10) (3,,) 1,)] )java.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 63
3 : [C2310 => Group( [ (), (), ( 6, 7, 8, 9,10)java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1
4 : [C2310 => Group( [ (), (11,12,13,14,15,16,17), ( 6, 7, 8, 9,10), (3,4,5),
  (1,2) ] )]
5 : [C2310 => Group( [ (20,21,22,23,24,25,26,27,28,29,30),
  (1,2,13,,1415,6,17),( 6,7 8,910) (3,4,5, (,2)]]
<span class=GAPprompt"gap>/span> <span class"GAPinput>HigherDimension PC5)<span>
6

</pre></div>

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