Title:

Rank 3 permutation groups with a regular normal subgroup

A (p ,n) group G is a permutation group (on a set Ω) which possesses a regular normal elementary abelian subgroup of order pn. The set Ω may be identified with a vector space V on which Go, the stabilizer of a point in G, acts as a subgroup of the general linear group GL(n,p). By a line of a subset ∆ of V, we mean the intersection of ∆ with a onedimensional subspace of V. The main result (Theorem 1.3.2) concerns (*)  groups, the term we give to rank 3 (p,n; groups in which the stabilizer of a point is doublytransitive on the lines of a suborbit. The essence or the problem is that of finding those subgroups of PGL (n,p) which have two orbits on the projective space PG (n – 1,p) and act doublytransitively on one of them. The notion of rank of a permutation group is discussed in 1.1, outline D.G. Higman’s combinatorial treatment of rank 3 groups. Associated with each permutation group having a regular subgroup is a certain S  ring, an algebraic structure which is basic to our theory. In 2.1 we define parameters of a rank 3 S  ring whd.ch coincide with those of any associated rank :3 group. Hence (*)  group with given parameters may be classified by finding all S  rings with the same parameters and then finding the associated (*)  groups. To assist in this task the concepts of residual Sring and the automorphism group of an Sring are introduced. Also of great value is Tamaschke’s notion of' the dual Sring, whi.ch is adapted to use in 2.2. In 3.1 we see how the imposition of conditions of transitivity on a suborbit of a rank 3 (p,n) groups leads to information about the parameters. In 3.3 the various relations connecting the parameters of' a (*) group are combined to yield specific sets of parameters, all of which are found in §4: to admit rank 3 S  rings. From results concerning the uniqueness of these S – rings, certain finite simple groups are characterised as their automorphism groups, and the proof of the main theorem is completed. A number of results are obtained as by – products in §4:, notably the answer to a question raised by Wielandt and a new representation of the simple group PSL(3,4) as a subgroup of PO(6,3, leading to an interesting presentation of a recentlydiscovered balanced block design. §5 is devoted to rank 3 (p,n) groups in which the transitivity condition on Go is replaced by the condition that the associated block design is balanced.
