Cyclic Steiner Triple Systems with Cyclic Subsystems

Abstract

A Steiner triple system of order v (briefly STS(v)) is a pair (V, B), where V is a v-set and B is a set of 3-subsets of V called triples such that each 2-subset of V is contained in exactly one triple. A cyclic Steiner triple system of order v (briefly CTS(v)), i.e. an STS with a regular cyclic automorphism group) can be thought of as a pair (Zv, B), where Zv is the additive group of integers modulo v fixing the set of triples B. An STS(v) exists iff v == lor 3 (mod 6), and a CTS(v) exists iff v == lor 3 (mod 6), v * 9 (see [5]). For a general reference on CTSs, see [2). When v = nu, n > 1, u > 1, then the group of integers modulo v will have a (non-trivial) subgroup of order u and index n for each divisor u of v. In a similar fashion, one can define a cyclic subsystem of a CTS(v) as having order u and index n, where (H, Bu) is the subsystem, and H is the subgroup of order u and index n which fixes Bu' A natural question asked, amongst others, by Peter Tannenbaum [6], is the following: What are the necessary and sufficient conditions for a CTS(v) to contain a cyclic subsystem of order u and index n? The problem becomes more interesting when one realizes that the obvious necessary conditions--namely that u divides v, and that u == lor 3 (mod 6), u * 9, v * 9--are not sufficient. First we consider construction of such systems. The methods we use are variants of known composition methods for cyclic designs. Recent references on such methods include Colbourn and Colbourn [1] and limbo and Kuriki [3]. Some cases of the main results of this paper are probable consequences of previously published work and, to an extent, may be considered folklore. However, for the purposes of clarity and simplicity, our presentation will be self-contained.