Conjugacy classes of maximal cyclic subgroups

In this paper, we study the number of conjugacy classes of maximal cyclic subgroups 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 precisely, we show that G / ⟨ G − ⟩ is either trivial, elementary abelian, a Frobenius group or isomorphic to A 5 .