On the Cameron-Praeger Conjecture

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 $P\text{GammaL}(2,p^e)$ with $p=2$ or 3, and $e$ is an odd prime power.