18.107 (2014)

Is there a finitely generated infinite residually finite $p$-group such that every subgroup of infinite index is cyclic?

The answer is known to be negative for $p = 2$. Note that in (M. Ershov, A. Jaikin Zapirain, J. Reine Angew. Math., 677 (2013), 71–134) it is shown that for every prime $p$ there is a finitely generated infinite residually finite $p$-group such that every finitely generated subgroup of infinite index is finite.