Conjugacy classes of maximal cyclic subgroups

In this paper, we study the number of conjugacy classes of maximal cyclic sub-groups of a finite group G, denoted n(G). First we consider the properties of this invariant in relation to direct and semi-direct products, and we characterize the normal subgroups N with (n)(G/N) = (n)(G). In addition, by applying the classification of finite groups whose nontrivial elements have prime order, we determine the structure of G/(G-), where G- is the set of elements of G generating non-maximal cyclic subgroups of G. More pre-cisely, we show that G/(G(-)) is either trivial, elementary abelian, a Frobenius group or isomorphic to A5.