Conjugacy classes of maximal cyclic subgroups

Abstract

Abstract In this paper, we study the number of conjugacy classes of maximal cyclic subgroups of a finite group 𝐺, denoted η ⁢ ( G ) \eta(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 η ⁢ ( G / N ) = η ⁢ ( G ) \eta(G/N)=\eta(G) . In addition, by applying the classification of finite groups whose nontrivial elements have prime order, we determine the structure of G / ⟨ G − ⟩ G/\langle G^{-}\rangle , where G − G^{-} is the set of elements of 𝐺 generating non-maximal cyclic subgroups of 𝐺. More precisely, we show that G / ⟨ G − ⟩ G/\langle G^{-}\rangle is either trivial, elementary abelian, a Frobenius group or isomorphic to A 5 A_{5} .