On finite 2-groups with relatively large centralizers of noninvariant subgroups. (Q2386438)

The authors consider finite 2-groups \(G\) of nilpotence class 2 in which \(|N_G(H):HC_G(H)|\leq 2\) for every non-normal subgroup \(H\). They show that in such a group the derived subgroup \(G'\) has order at most \(8\). Also, when \(G'\) has order precisely \(8\), then \(G'\) is elementary Abelian.