Automorphisms of Steiner Triple Systems

Abstract

This paper treats the following problem in combinatorial analysis: Find an incomplete balanced block design D with parameters b, v, r, k, and λ = 1. possessing an automorphism group G which is doubly transitive on the elements of D and such that the subgroup H of G fixing all the elements of a block is transitive on the remaining elements. Also find transitive extensions of such groups G. If the block design is a finite projective plane, the plane is necessarily Desarguesian. Thus, these properties of the automorphism group may be considered as a “Desarguesian” property of the designs. This paper considers the case in which D is a Steiner triple system. The main result is that a “Desarguesian” Steiner triple system is either 1) a projective geometry over GF(2) or 2) an affine geometry over GF(3). Two intermediate results are of interest: 1) A Steiner triple system has for each element an involution fixing only this element, if and only if every triangle generates an S(9), a Steiner triple system with 9 elements; 2) if a Steiner triple system has for each triple an involution fixing only the elements of this triple, then every triangle generates an S(7) or an S(9).