Conjugacy classes of maximal cyclic subgroups

Abstract

In this paper, we set $\eta (G)$ to be the number of conjugacy classes of maximal cyclic subgroups of $G$. We consider $\eta$ and direct and semi-direct products. We characterize the normal subgroups $N$ so that $\eta (G/N)=\eta (G)$. We set $G^{-}=\{g\in G\mid ⟨g⟩\mathrm{is}\mathrm{not}\mathrm{maximal}\mathrm{cyclic}\}$. We show if $⟨G^{-}⟩<G$, then $G/⟨G^{-}⟩$ is either (1) an elementary abelian $p$-group for some prime $p$, (2) a Frobenius group whose Frobenius kernel is a $p$-group of exponent $p$ and a Frobenius complement has order $q$ for distinct primes $p$ and $q$, or (3) isomorphic to $A_5$.

18 pages