19.91 (2018)

Let $G$ be a finite group with an abelian Sylow $p$-subgroup $A$. Suppose that $B$ is a strongly closed elementary abelian subgroup of $A$. Without invoking the Classification Theorem for Finite Simple Groups (CFSG), prove that $G$ has a normal subgroup $N$ such that $B = \Omega_1(A \cap N)$.

For $p = 2$, this is a corollary of a theorem of Goldschmidt. For $p$ odd, this has been proved by Flores and Foote (Adv. Math., 222 (2009), 453–484), but their proof relies on CFSG. A CFSG-free proof in the special case when $p = 3$ and $A$ has 3-rank 3 would already be quite interesting. This case arises in Aschbacher’s treatment of the $e(G) = 3$ problem, and a proof would provide an alternative to part of his argument. (For this application, one could assume that all proper simple sections of $G$ are known.)