### The Maxima on-line user's manual

Algebra Calculator

#### Todd_coxeter

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 variable `todd_coxeter_state`.

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```

```(%o1)                                true
(%i2) ```

? todd_coxeter;

Calculate

? todd_coxeter;

Calculate