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THE NUMBER OF CYCLIC SUBGROUPS OF FINITE ABELIAN GROUPS AND MENON’S IDENTITY

Part of: Elementary number theory Abelian groups Representation theory of groups

Published online by Cambridge University Press:  17 May 2019

MARIUS TĂRNĂUCEANU Open the ORCID record for MARIUS TĂRNĂUCEANU [Opens in a new window]
MARIUS TĂRNĂUCEANU*
Affiliation:
Faculty of Mathematics, “Al.I. Cuza” University, Iaşi, Romania email tarnauc@uaic.ro
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Abstract

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We give a new formula for the number of cyclic subgroups of a finite abelian group. This is based on Burnside’s lemma applied to the action of the power automorphism group. The resulting formula generalises Menon’s identity.

Keywords

Menon’s identityBurnside’s lemmagroup actioncyclic subgroupabelian grouppower automorphism group

MSC classification

Primary: 11A25: Arithmetic functions; related numbers; inversion formulas
Secondary: 20D60: Arithmetic and combinatorial problems 20K01: Finite abelian groups

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 101 , Issue 2 , April 2020 , pp. 201 - 206
Copyright
© 2019 Australian Mathematical Publishing Association Inc.

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