Abelian Steiner triple systems (original) (raw)

On the Szamkolowicz-Doyen Classification of Steiner Triple Systems

Proceedings of the London Mathematical Society, 1977

1. Introduction A Steiner triple system (8, stf) is a finite set 8 together with a family s/ of triples (subsets of cardinality 3) such that each pair of elements of S is contained in exactly one triple. The elements of 8 are called points, the elements of s/ are called lines, and three points which are not on a line are a triangle. The cardinal of 8, which we denote by 18 |, is the order of the Steiner triple system (S,s/). It is well known [8] that a necessary and sufficient condition for the existence of a Steiner triple system of order n > 0 is that n = 1 or 3 (mod 6). Trivial Steiner triple systems occur when S = stf -0, when 8 consists of one point and s& = 0, and when 8 consists of three points and $2 consists of one line. A subsystem of a Steiner triple system (S, stf) is a Steiner triple system (8', &4') such that 8' <= 8 and s#' <= s0. It is easy to see that if (8 V J^) , ..., (8 r ,s0 r ) are subsystems of a Steiner triple system then so is (8 x n... n8 r ,s^n... ns/ r ). It follows that given a set P s S of points, there is a unique minimal subsystem containing P. This subsystem is said to be generated by P.

Tricyclic Steiner Triple Systems

Graphs and Combinatorics, 2010

A Steiner triple system of order v, denoted ST S(v), is said to be tricyclic if it admits an automorphism whose disjoint cyclic decomposition consists of three cycles. In this paper we give necessary and sufficient conditions for the existence of a tricyclic ST S(v) for several cases. We also pose conjectures concerning their existence in two remaining cases. Keywords Steiner triple systems • Tricyclic automorphism 1 Introduction A Steiner triple system of order v, denoted ST S(v), is a v-element set, X , of points, together with a set β, of unordered triples of elements of X , called blocks, such that any two points of X are together in exactly one block of β. It is well known that a ST S(v) exists if and only if v ≡ 1 or 3 (mod 6). For a general review of triple systems in general, see [5]. An automorphism of a ST S is a permutation π of X which fixes β. A permutation π of a v-element set is said to be of type [π ] = [π 1 , π 2 ,. .. , π v

Linearly Derived Steiner Triple Systems

Designs, Codes and Cryptography - DCC, 1998

We attach a graph to every Steiner triple system. The chromatic number of this graph is related to the possibility of extending the triple system to a quadruple system. For example, the triple systems with chromatic number one are precisely the classical systems of points and lines of a projective geometry over the two-element field, the Hall triple systems have chromatic number three (and, as is well-known, are extendable) and all Steiner triple systems whose graph has chromatic number two are extendable. We also give a configurational characterization of the Hall triple systems in terms of mitres.

Some Remarks on Steiner Systems

2003

The main purpose of this paper is to introduce Steiner systems obtained from the finite classical generalized hexagons of order q. We show that we can take the blocks of the Steiner systems amongst the lines and the traces of the hexagon, and we prove some facts about the automorphism groups. Also, we make a remark concerning the geometric construction of a known class (KW) of Steiner systems and we deduce some properties of the automorphism group.

Pure Latin directed triple systems

Australas. J Comb., 2015

It is well-known that, given a Steiner triple system, a quasigroup can be formed by defining an operation · by the identities x · x = x and x·y = z where z is the third point in the block containing the pair {x,y}. The same is true for a Mendelsohn triple system where the pair (x,y )i s considered to be ordered. But it is not true in general for directed triple systems. However directed triple systems which form quasigroups under this operation do exist and we call these Latin directed triple systems. The quasigroups associated with Steiner and Mendelsohn triple systems satisfy the flexible law x · (y · x )=( x · y) · x but those associated with Latin directed triple systems need not. A directed triple system is said to be pure if when considered as a twofold triple system it contains no repeated blocks. In a previous paper, [Discrete Math. 312 (2012), 597– 607], we studied non-pure Latin directed triple systems. In this paper we turn our attention to pure non-flexible and pure flex...

Anti-mitre steiner triple systems

Graphs and Combinatorics, 1994

A mitre in a Steiner triple system is a set of five triples on seven points, in which two are disjoint. Recursive constructions for Steiner triple systems containing no mitre are developed, leading to such anti-mitre systems for at least ~ of the admissible orders. Computational results for small cyclic Steiner triple systems are also included.