Automorphism groups of Steiner triple systems

If $V$ is a Steiner triple system then there is an integer $N_V$ such that$,$ for $u\ge N_V$ and $u\equiv 1$ or $3$ $($mod$\,6),$ there is a Steiner triple system $U$ on $u$ points having $V$ as an ${\rm Aut} U$-invariant subsystem on which ${\rm Aut} U$ induces ${\rm Aut} V$ and ${\rm Aut} U\cong{\rm Aut} V$

Configurations of lines and models of Lie algebras

The automorphism groups of the 27 lines on the smooth cubic surface or the 28 bitangents to the general quartic plane curve are well-known to be closely related to the Weyl groups of $E\_6$ and $E\_7$. We show how classical subconfigurations of lines, such as double-sixes, triple systems or Steiner sets, are easily constructed from certain models of the exceptional Lie algebras. For ${\mathfrak e}\_7$ and ${\mathfrak e}\_8$ we are lead to beautiful models graded over the octonions, which display these algebras as plane projective geometries of subalgebras. We also interpret the group of the bitangents as a group of transformations of the triangles in the Fano plane, and show how this allows to realize the isomorphism $PSL(3,F\_2)\simeq PSL(2,F\_7)$ in terms of harmonic cubes.Comment: 31 page

Combinatorial designs and their automorphism groups

This thesis concerns the automorphism groups of Steiner triple systems and of cycle systems. Although most Steiner triple systems have trivial automorphism groups [2], it is widely known that for every abstract group, there exists a Steiner triple system whose automorphism is isomorphic to that group [16]. The well-known Bose construction [4] for Steiner triple systems, which has a number of variants, has a particularly nice structure, which makes it possible to say much about the automorphism group, and in the case of the construction based on an Abelian group, to derive the full automorphism group. The thesis contains a full analysis of these matters. Some of these results have been published by the author in [14]. The thesis also proves new results concerning the automorphism group for Steiner triple systems constructed using the tripling construction. An m-cycle system is a decomposition of a complete graph into cycles of length m. A Steiner triple system is thus a 3-cycle system. The thesis proves the result that for all m > 3, and for each abstract finite group, there exists an m-cycle system whose automorphism group is isomorphic to that group. In addition, the thesis contains a collection of new results concerning the conjecture by Furedi that every Steiner triple system is decomposable into triangles. Although this conjecture is expected to remain open for some time, it is possible to prove it for a number of standard constructions. It is further shown that for sufficiently large v, the number of Steiner triple systems of order v that are decomposable into triangles is at least vv2(1/54-0(1))

The 2-rotational Steiner triple systems of order 25

AbstractIn this paper, we enumerate the 2-rotational Steiner triple systems of order 25. There are exactly 140 pairwise non-isomorphic such designs. All these designs have full automorphism groups of order 12. We also investigate the existence of subsystems and quadrilaterals in these designs

A Pair of Disjoint 3-GDDs of type g^t u^1

Pairwise disjoint 3-GDDs can be used to construct some optimal constant-weight codes. We study the existence of a pair of disjoint 3-GDDs of type $g^t u^1$ and establish that its necessary conditions are also sufficient.Comment: Designs, Codes and Cryptography (to appear

Properties of Steiner triple systems of order 21

Properties of the 62,336,617 Steiner triple systems of order 21 with a non-trivial automorphism group are examined. In particular, there are 28 which have no parallel class, six that are 4-chromatic, five that are 3-balanced, 20 that avoid the mitre, 21 that avoid the crown, one that avoids the hexagon and two that avoid the prism. All systems contain the grid. None have a block intersection graph that is 3-existentially closed.Comment: 12 page

Distributive and anti-distributive Mendelsohn triple systems

We prove that the existence spectrum of Mendelsohn triple systems whose associated quasigroups satisfy distributivity corresponds to the Loeschian numbers, and provide some enumeration results. We do this by considering a description of the quasigroups in terms of commutative Moufang loops. In addition we provide constructions of Mendelsohn quasigroups that fail distributivity for as many combinations of elements as possible. These systems are analogues of Hall triple systems and anti-mitre Steiner triple systems respectively

Construction of Steiner quasigroups containing a specified number of subquasigroups of a given order

AbstractIn this paper we give a construction for Steiner quasigroups containing a specified number of subquasigroups of a given order. In particular, we show that, if there is a Steiner quasigroup of order v, v Steiner quasigroups of order q, where q > v, pairwise intersecting in the same quasigroup of order p, then, if q > vp and q − p is not divisible by the order of any non-trivial, proper subquasigroup of V there is a Steiner quasigroup of order v(q − p) + p containing a copy of each of the v quasigroups of order q and no other subquasigroups of order q

Schreier extensions of Steiner loops and extensions of Bol loops arising from Bol reflections

This dissertation explores two constructions of loop extensions: Schreier extensions of Steiner loops and a new extension formula for right Bol loops arising from Bol reflections.Steiner loops are a key tool in studying Steiner triple systems. We investigate extensions of Steiner loops, focusing in particular on the case of Schreier extensions, which provides a powerful method for constructing Steiner triple systems containing Veblen points. We determine the number of the Steiner triple systems of sizes 19, 27 and 31 with Veblen points, presenting some examples.Furthermore, we study a new extension formula for right Bol loops. We prove the necessary and sufficient conditions for the extension to be right Bol as well. We describe the most important invariants: right multiplication group, nuclei, center. We show that the core is an involutory quandle which is the disjoint union of two isomorphic involutory quandles. Lastly, we derive further results on the structure group of the core of the extension