Groups in Which the Normal Closures of Cyclic Subgroups Have Bounded Finite Hirsch–Zaitsev Rank

Abstract

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 is 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 r_hz(G). We study the groups in which the normal closure of every cyclic subgroup has the Hirsch–Zaitsev rank at most b (b is some positive integer). For some natural restrictions we find a function k₁(b) such that r_hz([G/Tor(G),G/Tor(G)]) ≤ k₁(b).