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 , λ ) designs with λ = 1 , except possibly when the group is PΓL ( 2 , p e ) with p = 2 or 3, and e is an odd prime power.