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In this paper, we study the number of conjugacy classes of maximal cyclic subgroups of a finite group 퐺, denoted η ⁢ (G) . First we consider the properties of this invariant in relation to direct and semi-direct products, and we characterize the normal subgroups 푁 with η ⁢ (G / N) = η ⁢ (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 퐺 generating non-maximal cyclic subgroups of 퐺. More precisely, we show that G / ⟨ G − ⟩ is either trivial, elementary abelian, a Frobenius group or isomorphic to A 5 . [ABSTRACT FROM AUTHOR]

Abstract: Let G be a finite group, and let denote the prime graph built on the set of conjugacy class sizes of G. In this paper, we consider the situation when has “few complete vertices”, and our aim is to investigate the influence of this property on the group structure of G. More precisely, assuming that there exists at most one vertex of that is adjacent to all the other vertices, we show that G is solvable with Fitting height at most 3 (the bound being the best possible). Moreover, if has no complete vertices, then G is a semidirect product of two abelian groups having coprime orders. Finally, we completely characterize the case when is a regular graph. [Copyright &y& Elsevier]