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14.4 (1999)

Open

a) Is it true that there exists a nilpotent group $G$ for which the lattice $\mathscr{L}(G)$ of all group topologies is not modular? (It is known that for abelian groups the lattice $\mathscr{L}(G)$ is modular and that there are groups for which this lattice is not modular: V. I. Arnautov, A. G. Topale, Izv. Akad. Nauk Moldova Mat., 1997, no. 1, 84–92 (Russian).)

b) Is it true that for every countable nilpotent non-abelian group $G$ the lattice $\mathscr{L}(G)$ of all group topologies is not modular?

Contributor: V. I. Arnautov

Progress

Editors' comment: (a) Yes, it is true (V. Arnautov, A. Topală, Bul. Acad. Ştiinţe. Repub. Moldova, Mat., 1998, no. 2(27), 130–131).
(b) No, there are such groups with modular $\mathscr{L}(G)$ (Dekui Peng, Preprint, 2023, https://arxiv.org/abs/2310.08269).

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