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Computational methods in the study
of permutation groups’
CHARLES C. SIMS
1. Introduction. One of the oldest problems in the theory of permutation
groups is the determination of the primitive groups of a given degree.
During a period of several decades a great deal of effort was spent on
constructing the permutation groups of low degree. For degrees 2 through
11 lists of all permutation groups appeared. For degrees 12 through 15
the lists were limited to the transitive groups, while only the primitive
groups of degrees 16 through 20 were determined. A detailed account of
the early work in this area can be found in the first article of [12] and references
to later papers are given in [1] and in [2] p. 564. The only recent
work of this type known to the author is that of Parker, Nikolai, and Appel
[13], [14], and a series of papers by Ito. These authors have shown that
for certain prime degrees any non-solvable transitive group contains the
alternating group.
The basic assumption of this paper is that it would be useful to extend
the determination of the primitive groups of low degree and that with
recent advances in group theory and the availability of electronic computers
for routine calculations it is feasible to carry out the determination
as far as degree 30 and probably farther. Ideally one would wish to have
an algorithm sufficiently mechanical and efficient to be carried out entirely
on a computer. Such an algorithm does not yet exist. The procedure outlined
in this paper combines the use of a computer with more conventional
techniques.
Suppose we wish to find the primitive groups of a given degree n. It
will be assumed that the primitive groups of degree less than n have already
been determined. Since a minimal normal subgroup of a primitive
group is transitive and is a direct product of isomorphic simple groups, the
first step is to take each of the known simple groups H, including the groups
of prime order, determine the transitive groups M of degree IZ isomorphic
to the direct product of one or more copies of H, and then for each
such group M find the primitive groups containing M as a normal subgroup.
While this is by no means a trivial procedure, it is considerably
12'
t This research was supported in part by the National Science Foundation.
169