COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

Some examples using coset enumeration
C.M. CAMPBELL
Introduction. A modification of the Todd-Coxeter coset enumeration
process [l] has been described by Campbell [2], Moser [3], and Benson and
Mendelsohn [4]. In this note we give some examples that illustrate the way
in which this modification is used.
Let G be an abstract group with a finite number of generators and relations,
and let H be a subgroup of G. Assume further that the index [G : HJ
of H in G is finite. Let E denote the identity and let (-) denote the inverse
of an element.
THIF&EM. Iffrom the relation R = E, where E is the identity and
we win the new information
R=al...a,...a,...a,, IGrGsep,
a.ap,+l . . . as = A
where each ai is a generator gj or its inverse and a, B are integers denoting
cosets, then
a.a, . . . a, = W./l,
where w= w,-, w,-, . . . WlW, . . . ws+l
is a word in the subgroup and a, /? are now thought of as coset representatives.
Proof. Express the relation R = E in the form
- - - -
a, . . . a, = ar-lar-2 . . . ala, . . . aS+l.
Then
- - ca.a,
. . . a, = a.a,-la,-2 . . . ala, . . . aS,l.
From previous information in the tables we find a-ii,-, expressed in the
form W,-,. y (a and y are now thought of as coset representatives and W,-,
is a word in the subgroup H):
a.a, . . . a, = W,-,y.4-2 . . . &Tip . . . iiS+l.
Now, again from the tables, y.&,_, = W,-,. 6.
Therefore
a.a, . . . a, = W,-lW,-26.(i,-3 . . . &ii, . . . &+l.
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