A note on polycylic residually finite-p groups

摘要

A subgroup H of a residually finite-p group G is almost p′-closed in G if H? has finite p-index in H, its closure with respect to the pro-p topology on G. We characterise polycyclic residually finite-p groups in which all subgroups are almost p-closed and discuss a few conditions that are sufficient for particular subgroups H to be almost p-closed. We also present, for each prime p, an example of a polycyclic residually-p group G for which |H?: H| takes on all possible values, including infinity, as H varies.