COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

176 Charles C. Sims
has a set of imprimitivity on Q-{u} of length m * 2, then H has an orbit
of length less than or equal to m - 1 on Q - {cc, p}.
Proof. Let d be a set of imprimitivity of G, with /d[ = m. Let
A = ((74 (6 4) I 6 * &,
A simple computation shows that
3 g E G such that I/g = a and (6, E}~ C d}.
1 A 1 = $z(n- i)(m- 1).
If we let r = {y 1 (y, {a, p}) E A}, then 1 r 1 = m- 1 and r is mapped
into itself by H.
If in Theorem 3.7 m s 4 and G* is primitive, then it follows from
Theorem 2.1 and the remark following it that G is abstractly a “known”
group.
The results of this section seem to indicate that at least for low degrees
most doubly transitive groups which are not doubly primitive will be sharply
doubly transitive or automorphism groups of block designs with L = 1.
The sharply doubly transitive groups are known. The question of which
block designs with 1 = 1 have doubly transitive .automorphism groups is
still open. It is known that projective planes with doubly transitive groups
are Desarguesian. Also, Hall [5] and Fischer [3] have made some progress
in the case of Steiner triple systems.
4. A computer program. In this section we present a short description of
a computer program for determining the order and some of the structure
of the group generated by a given set of permutations. Before describing
the program itself, however, it is necessary to discuss the method to be
used for storing a permutation group G in the computer. There are three
basic requirements which such a method should satisfy:
(1) it should be efficient with respect to storage,
(2) given a permutation h it should be easy to determine whether or
not h is in G,
(3) it is should be possible to run through the elements of G one at a
time without repetitions.
One method satisfying all of these conditions will now be presented.
Let G be a permutation group on G = (1, . . . , n}. Let G(O) = G and
for 1 < i G n- 1 let G(l) be the subgroup of G fixing 1, 2, . . . , i. Let uj
be a system of right coset representatives for GC’) in GC’-I), 1 s i =S n- 1.
Define
ni = 1 Vi/ = p-1): G(O[.
ni is the length of the orbit d of Go-l) containing i and for each j E d there
is a unique element in Ui taking i to j. Also
n-1
PI = pi
Computational methods for permutation groups
and every element of G has a unique representation of the form
&-l&-2 *. . g1,
where gi E Vi. We shall store the group G by storing the permutations in
each of the Ui. The total number of permutations stored is
n-1
I& Izi cl n(n+ 1)/2.
Thus the storage space required to store an arbitrary permutation group
of degree n grows as n3. By “packing” more than one integer into a computer
word, one can easily store a group of degree 50 on any of the large
computers available today. Because of the canonical form described above,
it is easy to run through the elements of G one at a time. Also, suppose
we are given a permutation h on Q and we wish to find .out if h is in G.
A necessary condition that h E G is that there exists gl E Ur such that hg;l
fixes 1. Similarly there must be a ga E UZ such that hg,-lg;l fixes 2. Continuing
in this manner we either arrive at elements gi E Ui such that
hgi’g,l . . . g,=‘l= 1
and so
h = gn-an-2 . . . gl E G,
or his not in G.
Permutation groups are not usually given by sets Ui and so we need a
program that will construct sets Ui for the group G generated by a set X
of permutations. Given X it is easy to construct U1. If for any g E G we
denote by d(g) the representative in U1 for the coset G@)g, then, by Lemmas
7.2.2 of [4], G(r) is generated by
x1 = {uxc#+4x)-lIu E u,, x E X}.
Continuing in this manner we can obtain generators for each of the
subgroups GC’) and construct sets Vi of coset representatives. There is one
difficulty which must be overcome. In general the set X1 will be much
larger than X. Unless some care is exercised, the sets of generators can
grow so large as to be unmanageable. This can be avoided by not constructing
all the generators of GC’) at one time, but rather as soon as a new
generator for GC’) has been obtained, using it to construct new elements
in Ui+l and new generators for G(‘+l). Also, whenever one of the elements
ux&ux)-l is computed, the process described above should be used to
determine ifit can be expressed in terms of the coset representative already
constructed and is therefore redundant.
A computer program of the type described has been written for the IBM
7040 at Rutgers. In its present form it can handle any group of degree 50
or less and can be of some use up to degree 127. Problem 5 on page 83 of
[4] can be done in slightly over a minute.
5. The primitive groups of degree not exceeding 20. In this section we
provide a list of the 129 primitive groups of degrees 2 to 20. The information
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