In a draft paper entitled "Some challenging group presentations"
George Havas, Derek F. Holt, P. E. Kenne and Sarah Rees say:

Our first example is a group which appeared in the Group-pub-forum
\cite{PUB} in 1996.  D.L. Johnson reported the failure of coset
enumeration, leaving a 1987 question of Malcom Wicks unanswered.

\begin{theorem} \mbox{}\\
The group $ W = \langle x, y \mid  x^3y^4x^5y^7 = 1 = x^2y^3x^7y^8 \rangle $
is cyclic of order $11$.
\end{theorem}

We believe that $W$ was first proved to be finite by P.E. Kenne, who
used a combination of Knuth-Bendix and Todd-Coxeter. As far as we know,
this was the first use of such a composite technique to solve a previously


used a combination of Knuth-Bendix and Todd-Coxeter. As far as we know,
this was the first use of such a composite technique to solve a previously
unsolved problem. For example, one
successful strategy was to use Knuth-Bendix to generate 500 new
relations in $W$, and then to use coset enumeration with respect to the
subgroup $\langle y \rangle$ using all of these relations. This
completed successfully, with the result $|W:\langle y \rangle| =1$,
after defining nearly 10 million cosets. From this it follows
immediately that $G$ is cyclic, and then we can deduce $|G|=11$ by
abelianising the presentation. So from the point of view of the first
successful solution of an open problem, this represents a success for
the combined approach.

However, further experiments by Kenne and others suggest that this
particular example is solved much more easily in terms of both time and
space by using Knuth-Bendix alone. Using a default shortlex ordering, it
completes in about 360 seconds (on a SparcStation 20) after finding about
26700 relations.  Our best result was obtained using a recursive-path
ordering with a length limit of 28 on the left and right hand sides of

stored equations, when it completed in 162 seconds after finding about
3250 equations.

In retrospect the Wicks group can be handled by coset enumeration alone. 
Using principles due to Mendelsohn \cite{Mend64} for relator 
table definitions (where cosets are applied to all cyclic permutations
of the relators and their inverses) we find that enumeration over
$\langle y \rangle$ gives index 1, defining less than a total of
22 million cosets with {\tt CT = 1000, RT = 100}. We are grateful to 
Colin Ramsay for pointing out the perhaps counter-intuitive result that
coset enumeration over the smaller subgroup $\langle x \rangle$ 
(that is, the trivial subgroup) is easier. With {\tt CT = 10, RT = 100} and
Mendelsohn-style definitions we obtain index 11 after defining a total
of about 14 million cosets.

Cheers,


George Havas               http://www.it.uq.edu.au/~havas
      Centre for Discrete Mathematics and Computing   
Department of Computer Science and Electrical Engineering  
 The University of Queensland, Queensland 4072 AUSTRALIA