On the regularity of irreducible subgroups of finite classical groups
Let $G$ be a finite group and let $\tau = (H_1, \ldots, H_t)$ be a $t$-tuple of core-free subgroups of $G$. We say that $\tau$ is regular if $G$ contains elements $g_1, \ldots, g_t$ such that $\bigcap_i H_i^{g_i} = 1$, which is equivalent to the existence of a regular $G$-orbit on the Cartesian product $G/H_1 \times \cdots \times G/H_t$. Regular tuples were first investigated by Anagnostopoulou-Merkouri and Burness in a paper from 2024, partly motivated by the aim of seeking a natural generalisation of the classical and widely studied concept of a base for a transitive permutation group, which aligns with the special case where the $H_i$ are pairwise conjugate subgroups. In this paper, we focus on the case where $G$ is a finite almost simple classical group and each $H_i$ is a maximal subgroup contained in Aschbacher's collection $\mathcal{S}$ of irreducibly embedded subgroups. Our main theorem determines all the non-regular $t$-tuples of this form with $t \geqslant 2$, which extends earlier work by Burness, Guralnick and Saxl in the base size setting. In particular, we deduce that every pair of maximal subgroups in $\mathcal{S}$ is regular if $n \geqslant 15$, where $n$ is the dimension of the natural module for the socle of $G$, and this lower bound is best possible.