1‐Rotational Steiner triple systems over arbitrary groups

Abstract

AbstractPhelps 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 (1981), 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