Abstract
Kirkman triple systems (KTSs) are among the most popular combinatorial designs and their existence has been settled a long time ago. Yet, in comparison with Steiner triple systems, little is known about their automorphism groups. In particular, there is no known congruence class representing the orders of a KTS with a number of automorphisms at least close to the number of points. We partially fill this gap by proving that whenever v equiv 39 (mod 72), or v equiv 4^e48 + 3 (mod 4^e96) and e ge 0, there exists a KTS on v points having at least v-3 automorphisms. This is only one of the consequences of an investigation on the KTSs with an automorphism group G acting sharply transitively on all but three points. Our methods are all constructive and yield KTSs which in many cases inherit some of the automorphisms of G, thus increasing the total number of symmetries. To obtain these results it was necessary to introduce new types of difference families (the doubly disjoint ones) and difference matrices (the splittable ones) which we believe are interesting by themselves.
Highlights
Steiner and Kirkman triple systems are undoubtedly amongst the most popular discrete structures
A Steiner triple system of order v, briefly STS(v), is a pair (V, B) where V is a set of v points and B is a set of 3-subsets of V with the property that any two distinct points are contained in exactly one block
A Kirkman triple system of order v, briefly Kirkman triple systems (KTSs)(v), is an STS(v) together with a resolution R of its block-set B, that is a partition of B into classes each of which is, in its turn, a partition of the point-set V
Summary
Steiner and Kirkman triple systems are undoubtedly amongst the most popular discrete structures. Adopting the combinatorial-analog of the famous Erlangen program by Klein [34], we believe that the interest of a discrete structure is proportional to the number of its automorphisms Motivated by this and by the shortage of results mentioned above, in this paper we deeply investigate Kirkman triple systems which are 3-pyramidal, i.e., admitting an automorphism group acting sharply transitively on all but three points. A STS(v) is called 1-transrotational if it has an automorphism group G that fixes exactly one point, switches two points, and acts sharply transitively on the remaining v − 3 This terminology was first used in [29] under the assumption that G is cyclic, though G just need to be binary. This will be shown in a paper in preparation [11] where we will deal with the case v ≡ 9 (mod 24)
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