Groups in which the normal closures of cyclic subgroups have bounded finite Hirsch–Zaitsev rank

In this paper, we study generalized soluble groups with restriction on normal closures of cyclic subgroups. A group $G$ is said to have finite Hirsch–Zaitsev rank if $G$ has an ascending series whose factors are either infinite cyclic or periodic and if the number of infinite cyclic factors are finite. It is not hard to see that the number of infinite cyclic factors in each of such series is an invariant of a group $G$. This invariant is called the Hirsch–Zaitsev rank of $G$ and will be denoted by $\mathbf r_{\mathrm{hz}}(G)$. We study the groups in which the normal closure of every cyclic subgroup has the Hirsch–Zaitsev rank at most $\mathbf b$ ($\mathbf b$ is some positive integer). For some natural restrictions we find a function $\mathbf k_1(\mathbf b)$ such that $\mathbf r_{\mathrm{hz}}([G/\mathrm{Tor}(G), G/\mathrm{Tor}(G)]) \leq \mathbf k_1(\mathbf b)$.