COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

172
Charles C. Sims
THEOREM 2.3. The matrices Mi, 1~ i< k, form a basis for a subalgebra
of the algebra of kX k matrices. In fact,
MiMj = Cm$viWk.
k
An r Xr matrix C will be called irreducible if there is no permutation D
of 1, . . ., r such that when 0 is applied to the rows and columns of C,
the result is a direct sum of two matrices. Because of the primitivity of
G we have
THEOREM 2.4. Fix i, 2 =z i == k, and let M = Mi and A = Bi. Then
(1) M and A are irreducible.
(2) M and A have the same minimal polynomial f.
(3) If we define vectors U,, q==O, by U, = (Q,. . .,u&, UO = (1, 0,
0, 0,. -.,O), Uq+l = U,M, then the trace of A4 is nu,,.
(4) ni is a root off and if ni = 81, 02,. . . , 8, are the distinct roots off,
then the multiplicity of 0, as an eigenvalue of A is
ej = trace fj(A)/fj(ej),
wherehfx) = f(x)/(x-ej)% and dj is the multiplicity of 0, in f. Also dl =
el = 1.
In view of part (3) of Theorem 2.4 the ej can be computed from a knowledge
of M. The fact that the ej must be positive integers imposes still
further conditions on the matrices Mi and the integers yti. Once matrices
Ml,. . ., Mk satisfying the conditions of Theorems 2.2, 2.3, and 2.4 have
been found, it is usually not particularly difficult to construct the possible
matrices B1,. . . , Bk, if they exist. This is of course equivalent to finding
the graphs $i as defined in [15]. Once the Bi are known, G must be a subgroup
of the group of permutation matrices commuting with each of the Bi.
3. Doubly transitive groups. In this section we take up the second of the
three cases described in the Introduction. Throughout, G will be assumed
to be doubly transitive on 9. For any subset d of Q, d* will denote the
set of 2-element subsets of d and G* will denote the permutation group
on .Q* induced by G. We shall not explicitly make the assumption that
G is not doubly primitive, but many of the results are trivial for doubly
primitive groups.
Of fundamental importance to the following discussion is the concept
of a block design. A block design, or more correctly a balanced incomplete
block design, with parameters ZJ, b, k, r, 3. is a set J’.J of points together with
a set B of blocks and an incidence relation between points and blocks such
that
(1) p = v,
(2) IBI = b,
(3) each block is incident with exactly k points,
Computational methods for permutation groups 173
(4) each point is incident with exactly r blocks,
(5) any two distinct points are incident with exactly A blocks.
The parameters of a block design satisfy the conditions
bk = rv, r(k- 1) = &v- 1).
A Steiner triple system is a block design with 1 = 1 and k = 3. A projective
plane is a block design with h = 1 and b = v 2 4. We shall say a block
design is trivial if for every k-element subset d of Q there is a block r such
that every point in d is incident with r. An automorphism of a block design
consists of a permutation g of the points and a permutation h of the
blocks such that a point a is incident with a block r if and only if c@ is
incident with rh. In many situations, in particular when 1 = 1, two blocks
are the same if they are incident with the same points. In this case h is
determined by g and we may consider the automorphism to be the permutation
g. For a more complete discussion of block designs the reader is
referred to [6]. We note that for any subset d of Q with jd 1 z= 2 the double
transitivity of G implies that (Q, B) is a block design, where
B= {dglgcG}
and incidence is set membership.
The following is an analogue of Theorem 18.2 of [16] for doubly transitive
groups.
THEOREM 3.1. Let G be a doubly transitive group on Q, let a and @ be
distinct points of Q, and let r be an orbit of Gors on Q-{a, ,8}. Then at
least one of the following holds:
(I) Every composition factor of Gti is a composition factor of some
subgroup of G$.
(2) G is a group of automorphisms of a non-trivial block design with
point set D for which 1, = 1.
Proof The following lemma is easily verified and we omit its proof.
LEMMA 3.2. Let A be a subset of Q with IAl s 2. Then A* is a set of
imprimitivity for G* if and only zf (Q, B) is a block design for which 2 = 1,
where
B = {Ag]gEG}.
Now assume that conclusion (2) of the theorem does not hold.
LEMMA 3.3. Let V + 1 be a subgroup of GJixing two points. There exists
gEG such that g-lVg = U< Gals and Ur =t= 1.
Proof Let y E r and let A be the fixed point set of V. 2 == IAl -K IQ/.
Consider
P = n Ag,
8