Conjugacy classes of maximal cyclic subgroups of metacyclic $p$-groups

DOI10.1515/JGTH-2022-0103arXiv2206.04339MaRDI QIDQ6401553

Rachel D. Camina, Mark L. Lewis, Mariagrazia Bianchi

Publication date: 9 June 2022

Abstract: In this paper, we set $eta(G)$ to be the number of conjugacy classes of maximal cyclic subgroups of a finite group $G$. We compute $eta(G)$ 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 $eta(G)gen-2$, and we determine when equality holds.

Mathematics Subject Classification ID

Conjugacy classes for groups (20E45) Arithmetic and combinatorial problems involving abstract finite groups (20D60) Finite nilpotent groups, (p)-groups (20D15)

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