COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

170 Charles C. Sims
easier than the second step, showing that there are no other primitive
groups of degree n. The list of primitive groups so far constructed can fail
to be complete only if there exists a primitive group G of degree n which
is a new simple group and so cannot be represented faithfully on fewer
than n points. Here it seems useful to distinguish three cases:
1. G is simply primitive, that is, primitive but not doubly transitive.
2. G is doubly transitive but not doubly primitive.
3. G is doubly primitive.
The first two cases will be dealt with in the next two sections. In the
third case, G is a transitive extension of one of the primitive groups of
degree n- 1 which contains only even permutations. The techniques of
transitive extension are discussed at length in [S] and [ll]. Thus it seems
necessary only to point out that the non-existence of transitive extensions
for several families of transitive groups is already known. See for example
[9], [10], and [16] p. 22. It seems probable that a computer program could
be written to construct up to isomorphism as permutation groups the
transition extensions of a given transitive group. So far, the author has
found that with the program presented in Q 4 of this paper and a small
amount of hand computation all transitive extensions can be determined
easily.
The paper concludes with a short description of a computer program
for finding the order and some of the structure of the group generated by
a given set of permutations and with a list of the 129 primitive groups
of degree not exceeding 20. This list was taken from the literature and
checked by the methods described here. The notation and terminology for
permutation groups is that of [16] with one important exception. The term
“block” is used only in the context of block designs and the older terms
“sets of imprimitivity” and “systems of imprimitivity” are used for what
are called blocks and complete block systems in [16]. Throughout G
will denote a primitive group on the finite set Q with jQ/ = n.
2. Simply primitive groups. In this section we discuss techniques for
handling the first of the three cases described in the Introduction. For
each acQ letdl(a) = {a},d~(a),. . ., ilk(a) be the orbits of G, numbered in
such a way that Oi(a’) = O,(a)” for all gE G. Let ni = [Ai(a We shall
assume 1 = nl==nz