2025/11/11 by Anagnostopoulou-Merkouri, Marina
#20B05 #FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.2511.08705
Let G\leqslant \rm Sym(Ω) be a finite transitive permutation group with point stabiliser H. A base for G is a subset of Ω whose pointwise stabiliser is trivial, and the minimal cardinality of a base is called the base size of G, denoted by b(G, Ω). Equivalently, b(G, Ω) is the minimal positive integer k such that G has a regular orbit on the Cartesian product Ωk. A well-known conjecture of Cameron from the 1990s asserts that if G is an almost simple primitive group and H is a so-called non-standard subgroup, then b(G, Ω) \leqslant 7, with equality if and only if G is the Mathieu group \rm M24 in its natural action of degree 24. This conjecture was settled in a series of papers by Burness et al. (2007-11). In this paper, we complete the proof of a natural generalisation of Cameron's conjecture. Our main result states that if G is an almost simple group and H1, …, Hk are any core-free non-standard maximal subgroups of G with k \geqslant 7, then G has a regular orbit on G/H1 × ⋯ × G/Hk, noting that Cameron's original conjecture corresponds to the special case where the Hi are pairwise conjugate subgroups. In addition, we show that the same conclusion holds with k \geqslant 6, unless G = \rm M24 and each Hi is isomorphic to \rm M23. For example, this means that if G is a simple exceptional group of Lie type and H1, …, H6 are proper subgroups of G, then there exist elements gi ∈ G such that \bigcapi Higi = 1. By applying recent work in a joint paper with Burness, we may assume G is a group of Lie type and our proof uses probabilistic methods based on fixed point ratio estimates.