Finite groups with small automizers of \(\mathcal{F}\)-subgroups (Q2400092)

The authors consider finite groups \(G\) satisfying the following natural condition that \(N_G(H)=HC_G(H)\) holds for every subgroup \(H\) of a group \(G\). The main result of the article is the following: Theorem. Let \(\mathcal{F}\) be a saturated formation such that the minimal non-\(\mathcal{F}\)-groups is soluble. Then, a finite group \(G\in\mathcal{F}\) iff \(H^{\mathcal{F}}\) satisfies the condition above for every subgroup \(H\) of \(G\).