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 $\mathcal{K}(S) \lhd G$.*



Status of a conjecture of Thompson
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.


