Abstract:
Let $G$ be a nilpotent group and $(\mathfrak M,<)$ be the lattice of all group topologies or the lattice of all group topologies in each of which the group $G$ possesses a basis of neighborhood of unit consisting of subgroups. If $\tau$ and $\tau'$ are group topologies from $\mathfrak M$ such that $\tau=\tau_0\prec_\mathfrak M\tau_1\prec_\mathfrak M\ldots\prec_\mathfrak M\tau_n=\tau'$, then $k\leq n$ for any chain $\tau=\tau'_0<\tau'_1<\ldots<\tau'_k=\tau'$ of topologies from $\mathfrak M$.
Keywords and phrases:topological groups, lattice of group topologies, modular lattice, chain of topologies, unrefinable chain.
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