COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

216 P. G. Ruud and R. Keown
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Some examples of man-machine interaction in
the solution of mathematical problems
N. S. MENDELSOHN
Summary. Three illustrative examples are given of how the enormous speed
and capacity of computing machines can be used to aid the mathematician in the
solution of problems he might not otherwise be willing to undertake. The ways in
which man and machine can interact are many and varied. The examples given
indicate three distinct directions in which such interplay can take place.
Example 1. The Sandler group. The collineation group of the free plane
generated by four points was studied by R. G. Sandler in [3]. The group has
its own intrinsic interest but there were two directions in which it appeared
that interesting information might be obtained. Two natural analogies suggested
themselves.
In the first case, by analogy with the situation in classical projective
planes, there was the possibility that this group, or at least a very large
subgroup, might yield a new simple group of very large order, and if this
were so one might expect an infinite class of such groups based on the
collineation groups of the free planes which are finitely generated.
Secondly, an analogy with group theory is possible. In group theory,
every group on k generators is a homomorphic image of the free group
with k generators. It might then be possible to show that the collineation
group of every projective plane generated by k points is a homomorphic
image of the collineation group of the free plane generated by four
points.
We show here that the second possibility is closer to the true situation.
Sandler’s group has the presentation
G= {A, B, C 1 A2 = B4 = Cz = (AB)3 = ((B2A)W)3 = CBAB”ACB2 = I}.
It can be shown that the element AC has infinite order. To study this
group it is convenient to look at homomorphic images in which AC is of
finite order. Accordingly, let G,, be the group obtained from Sandler’s
group by adjoining the relation (AC)” = I. It is not hard to see that A and
B generate the symmetric group S4 on four symbols, and that in any homomorphic
image of G the image of the subgroup generated by A and B must
be the full group S4 or the identity. Coset enumeration is used for the first
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