18.34 (2014)

Let $G$ be a topological group and $S = G_0 \leqslant \dots \leqslant G_n = G$ a subnormal series of closed subgroups. The infinite-length of $S$ is the cardinality of the set of infinite factors $G_i/G_{i-1}$. The virtual length of $G$ is the supremum of the infinite lengths taken over all such series of $G$. It is known that if $G$ is a pronilpotent group with finite virtual length then every closed subnormal subgroup is topologically finitely generated (N. Gavioli, V. Monti, C. M. Scoppola, J. Austral. Math. Soc., 95, no. 3 (2013), 343–355). Let $G$ be a pro-$p$ group (or more generally a pronilpotent group). If every closed subnormal subgroup of $G$ is topologically finitely generated, is it true that $G$ has finite virtual length?