19.93 (2018)

Conjecture: There exists a function $f: \mathbb{N} \times \mathbb{N} \to \mathbb{N}$ such that, if $G$ is a finite $p$-group with $d$ generators and $G$ has no epimorphic images isomorphic to the wreath product $C_p \wr C_p$, then each factor of the $p$-lower central series of $G$ has order bounded above by $f(p, d)$.

It is interesting to compare this conjecture with the celebrated characterization of Aner Shalev of finitely generated $p$-adic analytic pro-$p$-groups.