From: Derek Holt <dfh@maths.warwick.ac.uk>
Resent-Date:  Fri, 21 Jul 95 10:00:42 BST

Several years ago now, Heineken asked whether the group

  < x, y, z | z=[x,[x,y]],  x=[y,[y,z]],  y=[z,[z,x]] >

is finite or infinite. It was quickly discovered using the low index subgroup
and p-quotient algorithms, that it had a large finite perfect factor group of
structure  2^24.A5, but apart from that no progress was made.

A day or two ago Joachim Neubueser suggested that I try the Warwick Automatic
group programs on this example (which, for some reason, I appear never to
have done before), and to my surprise, they worked reasonably quickly
(the computation took about 15 minutes).
The conclusion is that the group is automatic with word-acceptor having
1106 states, and multipliers 2428 states.

This enables you to solve the word problem in the group, by reducing words to
their (shortlex) normal forms. From the word-acceptor, it can be seen 
immediately that the group is infinite - in fact, the generators have
infinite order.

Of course, it provides no further information on the the finite quotients,
so it is still unknown whether  2^24.A5 is the largest finite quotient group.

Derek Holt.