1-rotational Steiner triple systems over arbitrary groups

Phelps and Rosa introduced the concept of 1-rotational Steiner triple system, that is an STS(v) admitting an automorphism consisting of a fixed point and a single cycle of length v-1 [Discrete Math. 33 (1981), 57-66]. They proved that such an STS(v) exists if and only if v = 3 or 9 (mod 24). Here, we speak of a 1-rotational STS(v) in a more general sense. An STS(v) 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(v)'s by Phelps and Rosa are 1-rotational over the cyclic group. We denote by $A_{1r}$, $C_{1r}$, $Q_{1r}$, $G_{1r}$, the spectrum of values of v for which there exists a 1-rotational STS(v) over an abelian, a cyclic, a dicyclic, and an arbitrary group, respectively. In this paper, we determine $A_{1r}$ and find partial answers about $Q_{1r}$ and $G_{1r}$. 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 $SL_2(3)$, the special linear group of dimension 2 over $Z_3$.