COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

2 J. Neubiiser
is solved, the programme may be put out of use. There are a number of
problems tackled in this way, which we discuss in turn.
2.1. The construction of all groups of a particular kind,
2.1.1. A programme of this kind was first suggested as early as 1951
by M. H. A. Newman [Ne 4] for the investigation of the groups of order
256. P. Hall [Ha 7] had introduced the concept of isoclinism for the classification
and construction of p-groups. Newman pointed out that the number
of cases to be investigated for the determination of all groups of order
256 in a simple-minded use of Hall’s ideas would be far too big for computers
then (and would be even now). He gave an estimate to show that
by a probabilistic approach it would be feasible to obtain the great majority
of these groups in a reasonable time. It seems, however, that this
suggestion has never been followed.
2.1.2. In this conference C. C. Sims [Si 3] gave an outline of a different
procedure by which in principle each group of prime-power order would
be obtained just once. In this a group G of order p” is constructed as an
extension of the last term Qk(G) # ( 1 ) of its lower @-series, defined by
@r(G) = G, @i+r(G) = [G, @i(G)] ( gp 1 g E @i(G) ). The non-isomorphic
groups G with fixed OS,(G) 2 K and G/@,(G) Y H are in l-l correspondence
with those orbits of H2(H, K) under the joint action of both
automorphism groups A(H) and A(K), for which the extensions are groups
with @Q(G) = K. Sims has written a programme along these lines which
determined the two-generator groups of order 32 in a very short time.
According to him extrapolation from this experience would indicate quite
bearable computation times (‘v 10 hours) for the determination of most
of the groups of order 128. Special methods are probably necessary for
the case that His elementary abelian of order 32 and K elementary abelian
of order 4.
2.1.3. Also in this conference J. Cannon [Ca 31 reported that R. James
(Sydney) is determining the groups of order p6 for arbitrary primes p using
isoclinism. The calculations necessary in this set-up to find all non-isomorphic
groups in a given isoclinism class were done by a computer for the
first few primes and then generalized.
2.1.4. A listing of all primitive groups of low degrees is presently undertaken
by C. C. Sims [Si 2,4]. Earlier hand calculations went up to degree
20. These groups have been redetermined and the previous results found
correct. The calculations will be extended to higher degrees.
2.2. The Burnside problem. A recent survey of the problem is found in
[Ha 3], to which the reader is referred for definitions used and theoretical
results mentioned here. The finiteness of the Burnside groups B(n, r) of
exponent n on r generators is known for n = 2, 3, 4, 6 and all r, but the
order of B(n, r) is known only for n = 2, 3, 6, all r, and n = 4, r = 1, 2. In
Investigations of groups on computers 3
1962 M. Hall jr. [Ha l] outlined a programme by which in particular the
order of B(4,3) could be investigated. The idea was to use a Schreier
technique to find generators and relations for a suitable subgroup of B(4,3)
which could be handled. In 1964 M. Hall jr. and D. E. Knuth [Ha 2] announced
that with a programme applicable to arbitrary nilpotent Burnside
groups some results on B(4, 3) had been obtained, e.g. that the identity
(X,Y, z, w, w, w) = 1 holds modulo the seventh term of the lower central
series of this group. J. Leech [Le 2] has used Todd-Coxeterprogrammes (see
0 3.1) to obtain and to improve systems of defining relations for B(3, 3) and
B(4,2) and of groups of exponent 4 on three generators all or two of which
are of order 2. An investigation along different lines of the groups B(4, k) is
presently carried out by A. Tritter [Tr 11. He tries to prove that there is a
bound for the classes of the derived groups of the groups B(4, k), which
would be a consequence of a conjecture of G. Higman [Hi 11. For exponent
5 only the restricted Burnside problem has been solved. The biggest finite
group B*(5,2) of exponent 5 on two generators was found to be of class at
most 13 and order at most 534 . Recently, E. F. Krause and K. Weston
[Kr 3], starting from Kostrikin’s calculations, used a computer to establish
that these bounds are in fact attained.
Some of the programmes described in Q 3.1 may also give some information
on the restricted Burnside problem.
2.3. The search for simple groups. So far systematic searches with computers
have established only the non-existence of simple groups of certain
kinds.
2.3.1. In 1957 E. T. Parker and P. J. Nikolai [Pa 21 tried to find analogues
of the Mathieu groups Ml1 and Msa. Their computations showed that for
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