On \(p\)-nilpotence and \(IC\Phi\)-subgroups of finite groups (Q2064773)

The subgroup \(H\) of a finite group \(G\) is said to be an IC\(\Phi\)-subgroup of \(G\) if and only if \(H \cap [H, G] \leq \Phi(G)\), where \(\Phi(G)\) is the Frattini subgroup of \(G\). The author generalizes two theorems of \textit{Y. Gao} and \textit{X. Li} [Acta Math. Hung. 163, No. 1, 29--36 (2021; Zbl 1488.20019)] which give criteria for a group \(G\) to be \(p\)-nilpotent. Specifically, the author proves the following theorem.\par Let \(p\) be a prime dividing the order of a group \(G\) and let \(P\) be a Sylow \(p\)-subgroup of \(G\). Suppose that there is a subgroup \(D\) of \(P\) with \(1 < \vert D\vert \leq \vert P\vert \) such that any subgroup of \(P\) with order \(\vert D\vert \) is an IC\(\Phi\)-subgroup of \(G\). If \(\vert D\vert = 2\) and \(\vert P\vert \geq 8\), assume moreover that any cyclic subgroup of \(P\) with order 4 is an IC\(\Phi\)-subgroup of \(G\). Then \(G\) is \(p\)-nilpotent.\par The proof of this theorem and several lemmas are quite nice and proceed by looking at a counterexample of minimal order and obtaining a contradiction.