A negative answer to a question of Aschbacher

TL;DR

The paper provides infinitely many examples demonstrating that for simple groups, the lattice of overgroups can be a Boolean lattice of rank 2 with conjugate maximal overgroups, answering a question by Aschbacher.

Contribution

It constructs explicit examples showing a negative answer to Aschbacher's question about overgroup lattices in simple groups.

Findings

Existence of infinitely many such examples

Overgroup lattice can be Boolean of rank 2

Maximal overgroups can be conjugate in simple groups

Abstract

We give infinitely many examples to show that even for simple groups $G$ it is possible for the lattice of overgroups of a subgroup $H$ to be the Boolean lattice of rank $2$, in such a way that the two maximal overgroups of $H$ are conjugate in $G$. This answers negatively a question posed by Aschbacher.