arXiv subjects
Publications and source records attributed to R. D. Camina.
In this paper, we set $η(G)$ to be the number of conjugacy classes of maximal cyclic subgroups of a finite group $G$. We compute $η(G)$ for all metacyclic $p$-groups. We show that if $G$ is a metacyclic $p$-group of order $p^n$ that is not dihedral, generalized quaternion, or semi-dihedral, then $η(G) \ge n-2$, and we determine when equality holds.
In 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 $A_5$.
In this paper, we set $η(G)$ to be the number of conjugacy classes of maximal cyclic subgroups of $G$. We prove that if $G$ is a $p$-group of order $p^n$ and nilpotence class $l$, then $η(G)$ is bounded below by a linear function in $n/l$.