The nilpotency class of \(p\)-groups in which subgroups have few conjugates. (Q2496860)

The main result is the following assertion: Let \(G\) be a finite \(p\)-group. If \(|G:N_G(C)|\leq p^2\) for all cyclic \(C\leq G\), then \(\text{cl}(G)\leq 4\). This result is best possible as shows, for every \(p\), any \(p\)-group of maximal class, of order \(p^5\) and exponent \(p^2\). The paper contains a number of results presenting independent interest.