1‐Rotational Steiner triple systems over arbitrary groups

Abstract

Phelps and Rosa introduced the concept of 1-rotational Steiner triple system, that is an STS(ν) admitting an automorphism consisting of a fixed point and a single cycle of length ν − 1 [Discrete Math. 33 (12), 57–66]. They proved that such an STS(ν) exists if and only if ν ≡ 3 or 9 (mod 24). Here, we speak of a 1-rotational STS(ν) in a more general sense. An STS(ν) is 1-rotational over a group G when it admits G as an automorphism group, fixing one point and acting regularly on the other points. Thus the STS(ν)'s by Phelps and Rosa are 1-rotational over the cyclic group. We denote by 𝒜1r, 𝒞1r, 𝒬1r, 𝒢1r, the spectrum of values of ν for which there exists a 1-rotational STS(ν) over an abelian, a cyclic, a dicyclic, and an arbitrary group, respectively. In this paper, we determine 𝒜1r and find partial answers about 𝒬1r and 𝒢1r. The smallest 1-rotational STSs have orders 9, 19, 25 and are unique up to isomorphism. In particular, the only 1-rotational STS(25) is over SL2(3), the special linear group of dimension 2 over Z3. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 215–226, 2001