Conjugacy classes of maximal cyclic subgroups of metacyclic 𝑝-groups-Reference-Cited by-同舟云学术

Conjugacy classes of maximal cyclic subgroups of metacyclic 𝑝-groups

Published:2022-10-15 Issue:0 Volume:0 Page:
ISSN:1433-5883
Container-title:Journal of Group Theory
language:en
Short-container-title:

Author:

Bianchi Mariagrazia¹,Camina Rachel²,Lewis Mark L.³^ORCID

Affiliation:

1. Dipartimento di Matematica F. Enriques , Università degli Studi di Milano , via Saldini 50, 20133 Milano , Italy

2. Fitzwilliam College , Cambridge , CB3 0DG , United Kingdom

3. Department of Mathematical Sciences , Kent State University , Kent , Ohio, 44242 USA

Abstract

Abstract In this paper, we set η ⁢ ( G )

\eta(G)

to be the number of conjugacy classes of maximal cyclic subgroups of a finite group 𝐺. We compute η ⁢ ( G )

\eta(G)

for all metacyclic 𝑝-groups. We show that if 𝐺 is a metacyclic 𝑝-group of order p n

p^{n}

that is not dihedral, generalized quaternion, or semi-dihedral, then η ⁢ ( G ) ≥ n - 2

\eta(G)\geq n-2

, and we determine when equality holds.

Publisher

Walter de Gruyter GmbH

Subject

Algebra and Number Theory

Link

https://www.degruyter.com/document/doi/10.1515/jgth-2022-0103/pdf

Reference9 articles.

1. Y. Barnea, R. D. Camina, M. Ershov and M. L. Lewis, On groups that can be covered by conjugates of finitely many cyclic or procyclic subgroups, preprint.

2. J. R. Beuerle, An elementary classification of finite metacyclic 𝑝-groups of class at least 3, Algebra Colloq. 12 (2005), no. 4, 553–562.

3. M. Bianchi, R. D. Camina and M. L. Lewis, Conjugacy classes of maximal cyclic subgroups and nilpotence class of 𝑝-groups, Bull. Aust. Math. Soc. (2022), 10.1017/S0004972722000211.

4. M. Bianchi, R. D. Camina, M. L. Lewis and E. Pacifici, Conjugacy classes of maximal cyclic subgroups, preprint (2022), https://arxiv.org/abs/2201.05637.

5. J. D. Dixon, M. P. F. du Sautoy, A. Mann and D. Segal, Analytic Pro-p-Groups, London Math. Soc. Lecture Note Ser. 157, Cambridge University, Cambridge, 1991.

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