Conjugacy classes of maximal cyclic subgroupsIn this paper, we set $η(G)$ to be the number of conjugacy classes of
maximal cyclic subgroups of $G$. We consider $η$ and direct and semi-direct
products. We characterize the normal subgroups $N$ so that $η(G/N) = η
(G)$. We set $G^- = \{ g \in G \mid \langle g \rangle {\rm ~is~not
~maximal~cyclic} \}$. We show if $\langle G^- \rangle < G$, then $G/\langle G^-
\rangle$ 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 $A_5$.
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