20.121 (2022)

Let $G$ be a finite almost simple group such that $\text{Soc}(G)$ is a non-soluble group of Lie type over a field of characteristic $p$. Let $R$ be a Sylow $q$-subgroup of $G$ (for some $q$) and let $\text{Min}_G(R)$ be the subgroup of $R$ generated by all minimal by inclusion intersections of the form $R \cap R^g$, where $g \in G$. Is it true that for $p > 3$ the subgroup $\text{Min}_G(R)$ is non-trivial if and only if the following hold: $G = \text{Aut}(L_2(p))$, where $p$ is a Mersenne prime, $\text{Min}_G(R) = R$, and $q = 2$.

It is known that this is not always the case for $p = 2, 3$.