Status of a conjecture of Thompson

Question

Let $ S $ be a finite group. Denote by $\mathcal{B}_0(S)$ the set of the subgroups $H$ of $S$ satisfying $|H:H'| > |K:K'|$ for every proper subgroup $K$ of $H$ ($H'$ denotes the drived subgroup of $H$), and let $\mathcal{K}(S)$ be the subgroup generated by the minimal elements of $\mathcal{B}_0(S)$ (the latter being ordered by inclusion). Is the following conjecture of Thompson ["A Replacement theorem for p-groups and a Conjecture" J. Algebra 13 (1969)] still open?

Conjecture. Let $G$ be a finite p-solvable group which doesn't involve $\mathrm{SL}_2(p)$ , and $S$ is a p-sylow of $G$. If $\mathrm{O}_{p'}(G) = 1$, then $\mathcal{K}(S) \lhd G$.

Answer 1

The conjecture is still open. The most recent paper where this Thompson's paper was mentioned was published in 2016: Rowley, Peter; Taylor, Paul An algorithm for the Thompson subgroup of a p-group. J. Algebra 461 (2016), 375–389.

There is also a more recent paper in the arXive: An extension of the Glauberman ZJ-Theorem by Yasir Kizmaz.

It looks like the largest class of finite groups where Thompson's idea was used was the class of $p$-groups. For that class the conjecture was proved by J. Thompson himself.