Steiner Triple Systems with an Involution

Let π be a permutation of the set {1, 2,..., υ} having f< υ fixed points and (υ — f)/2 disjoint transpositions. We investigate the existence of Steiner triple systems admitting π as an auto-morphism. When f = 1 such a system is known as a reverse Steiner triple system and it is known that reverse Steiner triple systems exist if and only if υ ≡ 1, 3, 9 or 19 (mod 24). In this paper we show that a Steiner triple system admitting π as an automorphism, and f > 1 exists if and only if υ ≡ 1 or 3(mod 6), f ≡ 1 or 3(mod 6), and either (υ — f ≡ 0(mod 4), and υ ⩾ 2f + 1) or (υ — f ≡ 2 (mod 4), and υ ⩾ 3f).