Some progress on the existence of 1-rotational Steiner triple systems |
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Authors: | Simona Bonvicini Marco Buratti Gloria Rinaldi Tommaso Traetta |
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Institution: | (1) Dipartimento di Matematica e Informatica, Università di Perugia, Via Vanvitelli 1, 06123 Perugia, Italy |
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Abstract: | 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 a 1-rotational STS(v) under a cyclic, an abelian, or a dicyclic group, has been established in Phelps and Rosa (Discrete Math 33:57–66, 1981), Buratti (J Combin Des 9:215–226, 2001) and Mishima (Discrete Math 308:2617–2619, 2008), respectively. Nevertheless, the spectrum of values of v for which there exists a 1-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 \not o 0{n \not\equiv 0} (mod 4), and the odd part of (p
3 − p)n that is square-free and without prime factors congruent to 1 (mod 6). |
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