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16.103 (2006)

Solved

Is there a rank analogue of the Leedham-Green–McKay–Shepherd theorem on $p$-groups of maximal class? More precisely, suppose that $P$ is a 2-generator finite $p$-group whose lower central quotients $\gamma_i(P)/\gamma_{i+1}(P)$ are cyclic for all $i \geqslant 2$. Is it true that $P$ contains a normal subgroup $N$ of nilpotency class $\leqslant 2$ such that the rank of $P/N$ is bounded in terms of $p$ only?

Contributor: E. I. Khukhro

Progress

No, moreover, there are no functions $d(p)$ and $r(p)$ such that a group with these properties would necessarily have a normal subgroup of derived length $\leqslant d(p)$ with quotient of rank $\leqslant r(p)$ (E. I. Khukhro, Siber. Math. J., 54, no. 1 (2013), 174–184).

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