Some progress on the existence of 1-rotational Steiner Triple Systems

A Steiner Triple System of order v (briefly STS(v)) is 1-rotational under G if it admits G as an automorphism group acting sharply transitively on all but one point.The spectrum of values of v for which there exists a1-rotational STS(v) under a cyclic, an abelian, or a generalized quaternion group, has beenestablished in 1981 (phelps and Rosa), in 2001 (Buratti) and in 2008 (Mishima), respectively.Nevertheless, the spectrum of values of v for which there exists a1-rotational STS(v) under an arbitrary group has not been completely determined yet.This paper is a considerable step forward to the solution of this problem.In fact, we leave as uncertain cases only those for which we have v = (p^3-p)n + 1 = 1 (mod 96)with p a prime, n =1,2,3 mod 4, and the odd part of (p^3-p)n that is square-free and without prime factors congruent to 1 mod 6.