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THE SECOND MINIMUM/MAXIMUM VALUE OF THE NUMBER OF CYCLIC SUBGROUPS OF FINITE $p$-GROUPS

Part of: Representation theory of groups

Published online by Cambridge University Press:  20 April 2020

MIHAI-SILVIU LAZOREC Open the ORCID record for MIHAI-SILVIU LAZOREC [Opens in a new window] ,
RULIN SHEN Open the ORCID record for RULIN SHEN [Opens in a new window]  and
MARIUS TĂRNĂUCEANU Open the ORCID record for MARIUS TĂRNĂUCEANU [Opens in a new window]
MIHAI-SILVIU LAZOREC*
Affiliation:
Faculty of Mathematics, ‘Al.I. Cuza’ University, Iaşi, Romania email mihai.lazorec@student.uaic.ro
RULIN SHEN
Affiliation:
Department of Mathematics, Hubei Minzu University, Enshi, Hubei, PR China email rshen@hbmy.edu.cn
MARIUS TĂRNĂUCEANU
Affiliation:
Faculty of Mathematics, ‘Al.I. Cuza’ University, Iaşi, Romania email tarnauc@uaic.ro
*
✉mihai.lazorec@student.uaic.ro
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Abstract

Let $C(G)$ be the poset of cyclic subgroups of a finite group $G$ and let $\mathscr{P}$ be the class of $p$-groups of order $p^{n}$ ($n\geq 3$). Consider the function $\unicode[STIX]{x1D6FC}:\mathscr{P}\longrightarrow (0,1]$ given by $\unicode[STIX]{x1D6FC}(G)=|C(G)|/|G|$. In this paper, we determine the second minimum value of $\unicode[STIX]{x1D6FC}$, as well as the corresponding minimum points. Since the problem of finding the second maximum value of $\unicode[STIX]{x1D6FC}$ has been solved for $p=2$, we focus on the case of odd primes in determining the second maximum.

Keywords

number of cyclic subgroupsfinite p-groupsexponent of a group

MSC classification

Primary: 20D60: Arithmetic and combinatorial problems
Secondary: 20D15: Nilpotent groups, $p$-groups 20D25: Special subgroups (Frattini, Fitting, etc.)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 103 , Issue 1 , February 2021 , pp. 96 - 103
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

The first author was supported by the European Social Fund, through Operational Programme Human Capital 2014-2020, Project No. POCU/380/6/13/123623. The second author was supported by NSF of China, Grant No. 11561021.

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