On a characterization of quasicyclic groups

Abstract

Let G be an infinite solvable group (resp. an infinite group properly containing its commutator subgroup G ' ). We prove that G is isomorphic to a quasicyclic group if and only if all proper normal subgroups of G are finitely generated (resp. all proper normal subgroups of G are cyclic or finite). In this paper, the symbols Q; Z; N denote the rational numbers, the integers, the nonnegative integers, respectively. A quasicyclic group (or Prufer group) is the p-primary component of Q/Z, that is, the unique maximal p-subgroup of Q/Z, for some prime number p. Any group isomorphic to it will also be called a quasicyclic group and denoted by Zp1. Quasicyclic groups play an important roles in the infinite abelian group theory. They may also be defined in a number of equivalent ways (again, up to isomorphism): • A quasicyclic group is the group of all p n -th complex roots of 1, for