Intersection matrices for finite permutation groups

In this paper we study finite transitive groups G acting on a set Q. The results, which are trivial for multiply-transitive groups, directly generalize parts of the discussion of rank-3 groups in [4] and [5l. There arc close connections with Feit and Higman’s paper [2]. For each a EQ let us choose a G,-orbit d(a) # {CZ} so that d(a)8 = d(a”) for all a E Sz and g E G. Relative to LI we introduce a distunxe in Q based on taking the points of d(u) to be at distance 1 from a (see Section 1). The maximum distance we call the diameter of G. A necessary and sufficient condition for G to be primitive is that the diameter be finite with respect to every d. It is important to note, however, that finiteness of the diameter with respect to a single d does not imply primitivity. We study the matrix M of intersection numbers of A (defined in Section 4). M is irreducible if and only if G has finite diameter with respect to A, and in this case the subdegrees and diameter are determined by M. The minimum polynomial of M is shown to coincide with that of the incidence matrix A of d, and it is shown how to compute the trace of A*, q 3 0, in terms of M. This means that if p(x) is a polynomial such that p(M) = 0 and if 0 is a root of p(x), then the multiplicity of 0 as an eigenvalue of A is determined by M. In case the minimum and characteristic polynomials of M coincide, we show that M has simple eigenvalues, from which it follows that the irreducible constituents of the permutation representation have multiplicity 1 and that the degrees of these constituents are determined by M. In this case there is a one-to-one correspondence between the eigenvalues of M and the irreducible constituents of the permutation representation, which preserves conjugacy. The new simple group of order 750, 560 discovered by Janko [6] provides an example in which two of the constituents are conjugate even though the subdegrces are distinct.