An orbit theorem for Steiner triple systems

✍ Scribed by Peter J. Cameron

• Publisher: Elsevier Science
• Year: 1994
• Tongue: English
• Weight: 206 KB
• Volume: 125
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(94)90148-1

✦ Synopsis

An automorphism

group of a nontrivial (possibly infinite) Steiner triple system has at least as many block-orbits as point-orbits.

The proof is a translation of that in the finite case, with a small twist. For block size 4, the argument fails, and, indeed, the stronger statement (involving block-tactical decompositions) is false.

An orbit theorem for an incidence structure (X,6@ is typically the assertion that any automorphism group has at least as many orbits on the block set 29 as on the point set X. I will briefly consider how such a theorem is proved for finite structures, in order to extend it to infinite ones.

Let Rx and IwB denote the real vector spaces of real-valued functions on points and blocks, respectively. The incidence transformation is the map 1: [w*+lR~ defined by

for XEX, BE$~. It is the transformation whose matrix, relative to the bases consisting of characteristic functions of singletons, is the incidence matrix of the structure.

If G is a group acting on X, then the dimension of the space Fixc(Rx) of fixed functions for G in Rx is equal to the number of G-orbits in X -a basis for Fix&Rx) consists of the characteristic functions of the G-orbits. If G is a group of automorphisms of (X, @), then 1 maps Fix&RX) into Fix,@"). Hence, we have the following proposition.