Steiner Triple Systems with Block-Transitive Automorphism Groups

Citation

Clapham, Paul Charles (1974) Steiner Triple Systems with Block-Transitive Automorphism Groups. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/8FYV-1J76. https://resolver.caltech.edu/CaltechETD:etd-10172005-105806

Abstract

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. If G is an automorphism group of a Steiner triple system which is doubly transitive on the points, then it is transitive on the blocks. It is shown that the converse is false and that all counterexamples have odd order. All Steiner triple systems which have a block-transitive but not doubly point-transitive group of automorphisms are described. They include the Euclidean geometries of odd dimension over GF(3), a class of systems first described by Netto in 1893, and another class of systems. A system in this third class has a group of automorphisms acting regularly on the blocks, and the number of points is a prime power congruent to 7 modulo 12. The number of such systems (up to isomorphism) with a prime number of points p, where [...] (mod 12), is shown to be in the interval [...]. The classification of block-transitive Steiner triple systems is applied to prove the following theorem: if G is a doubly transitive automorphism group of a Steiner triple system and P is a p-subgroup of G maximal subject to the condition that it fix more than three points, then the points fixed by P form a subsystem with a doubly transitive automorphism group.

Thesis Files