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GROUPS HAVING 12 CYCLIC SUBGROUPS

Part of: Representation theory of groups

Published online by Cambridge University Press:  26 August 2025

KHYATI SHARMA Open the ORCID record for KHYATI SHARMA [Opens in a new window]  and
A. SATYANARAYANA REDDY Open the ORCID record for A. SATYANARAYANA REDDY [Opens in a new window]
KHYATI SHARMA*
Affiliation:
Department of Mathematics, Shiv Nadar Institution of Eminence , NH-91, Dadri, Gautam Buddha Nagar, Uttar Pradesh 201314, India
A. SATYANARAYANA REDDY
Affiliation:
Department of Mathematics, Shiv Nadar Institution of Eminence , NH-91, Dadri, Gautam Buddha Nagar, Uttar Pradesh 201314, India e-mail: satya.a@snu.edu.in
*
e-mail: khyatisharma0907@gmail.com
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Abstract

A finite group is said to be n-cyclic if it contains n cyclic subgroups. For a finite group G, the ratio of the number of cyclic subgroups to the number of subgroups is known as the cyclicity degree of the group G and is denoted by $\textit {cdeg} (G)$. In this paper, we classify all $12$-cyclic groups. We also prove that the set of cyclicity degrees for all the finite groups is dense in $[0,1]$, which solves a problem posed by Tărnăuceanu and Tóth [‘Cyclicity degrees of finite groups’, Acta Math. Hungar. 145(2) (2015), 489–504].

Keywords

n-cyclic groupnumber of subgroupsnumber of cyclic subgroupsSylow’s theoremsDedekind group

MSC classification

Primary: 20D60: Arithmetic and combinatorial problems
Secondary: 20D25: Special subgroups (Frattini, Fitting, etc.) 20D20: Sylow subgroups, Sylow properties, $pi$-groups, $pi$-structure

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , First View , pp. 1 - 12
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

The first author would like to acknowledge the UGC-SRF grant, which is enabling her doctoral work.

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