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<h1 class="chapter"> 34. Groups </h1>
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<tr><td align="left" valign="top"><a href="#SEC133">34.1 Definitions for Groups</a></td><td> </td><td align="left" valign="top">
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<h2 class="section"> 34.1 Definitions for Groups </h2>
<dl>
<dt><u>Function:</u> <b>todd_coxeter</b><i> (<var>relations</var>, <var>subgroup</var>)</i>
<a name="IDX1125"></a>
</dt>
<dt><u>Function:</u> <b>todd_coxeter</b><i> (<var>relations</var>)</i>
<a name="IDX1126"></a>
</dt>
<dd><p>Find the order of G/H where G is the Free Group modulo <var>relations</var>, and
H is the subgroup of G generated by <var>subgroup</var>. <var>subgroup</var> is an optional
argument, defaulting to []. In doing this it produces a
multiplication table for the right action of G on G/H, where the
cosets are enumerated [H,Hg2,Hg3,...]. This can be seen internally in
the <code>$todd_coxeter_state</code>.
</p>
<p>The multiplication tables for the variables are in
<code>table:todd_coxeter_state[2]</code>. Then <code>table[i]</code> gives the table for
the ith variable. <code>mulcoset(coset,i) := table[varnum][coset];</code>
</p>
<p>Example:
</p>
<table><tr><td> </td><td><pre class="example">(%i1) symet(n):=create_list(
if (j - i) = 1 then (p(i,j))^^3 else
if (not i = j) then (p(i,j))^^2 else
p(i,i) , j, 1, n-1, i, 1, j);
<3>
(%o1) symet(n) := create_list(if j - i = 1 then p(i, j)
<2>
else (if not i = j then p(i, j) else p(i, i)), j, 1, n - 1,
i, 1, j)
(%i2) p(i,j) := concat(x,i).concat(x,j);
(%o2) p(i, j) := concat(x, i) . concat(x, j)
(%i3) symet(5);
<2> <3> <2> <2> <3>
(%o3) [x1 , (x1 . x2) , x2 , (x1 . x3) , (x2 . x3) ,
<2> <2> <2> <3> <2>
x3 , (x1 . x4) , (x2 . x4) , (x3 . x4) , x4 ]
(%i4) todd_coxeter(%o3);
Rows tried 426
(%o4) 120
(%i5) todd_coxeter(%o3,[x1]);
Rows tried 213
(%o5) 60
(%i6) todd_coxeter(%o3,[x1,x2]);
Rows tried 71
(%o6) 20
(%i7) table:todd_coxeter_state[2]$
(%i8) table[1];
(%o8) {Array: (SIGNED-BYTE 30) #(0 2 1 3 7 6 5 4 8 11 17 9 12 14 #
13 20 16 10 18 19 15 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0)}
</pre></td></tr></table>
<p>Note only the elements 1 thru 20 of this array <code>%o8</code> are meaningful.
<code>table[1][4] = 7</code> indicates coset4.var1 = coset7
</p>
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