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An $N^0$-group is a group in which every cyclic subgroup is a term of some normal system of the group.

Is every $N^0$-group an $\tilde{N}$-group?

Contributor: Sh. S. Kemkhadze

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No. Every group $G$ with a central system $\{H_i\}$ is an $N^0$-group, since for any $g \in G$ the system $\{\langle g, H_i \rangle\}$ refined by the trivial subgroup and the intersections of all the subsystems is a normal system of $G$ containing $\langle g \rangle$. Free groups have central systems but are not $\tilde{N}$-groups. (Yu. I. Merzlyakov, 1973.)

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