On the Cameron-Praeger Conjecture

TL;DR

This paper proves the Cameron-Praeger conjecture for non-trivial Steiner 6-designs, confirming no such block-transitive designs exist except possibly under specific group conditions.

Contribution

It confirms the conjecture for Steiner 6-designs, narrowing down the cases where non-trivial block-transitive 6-designs could exist.

Findings

Cameron-Praeger conjecture is true for Steiner 6-designs

Exceptions may occur with groups PΓL(2,p^e) for p=2 or 3, e odd prime

No non-trivial block-transitive 6-designs found outside these cases

Abstract

This paper takes a significant step towards confirming a long-standing and far-reaching conjecture of Peter J. Cameron and Cheryl E. Praeger. They conjectured in 1993 that there are no non-trivial block-transitive 6-designs. We prove that the Cameron-Praeger conjecture is true for the important case of non-trivial Steiner 6-designs, i.e. for 6-$(v,k,\lambda)$ designs with $\lambda=1$, except possibly when the group is $P\GammaL(2,p^e)$ with $p=2$ or 3, and $e$ is an odd prime power.