Functions and Variables for Groups
==================================

 -- Function: todd_coxeter (RELATIONS, SUBGROUP)
 -- Function: todd_coxeter (RELATIONS)

     Find the order of G/H where G is the Free Group modulo RELATIONS, and H is the subgroup of G generated by SUBGROUP.  SUBGROUP 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 $todd_coxeter_state.

     The multiplication tables for the variables are in table:todd_coxeter_state[2].  Then table[i] gives the table for the ith variable.  mulcoset(coset,i) := table[varnum][coset];

     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)}

     Note only the elements 1 thru 20 of this array %o8 are meaningful. table[1][4] = 7 indicates coset4.var1 = coset7