Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-18T03:08:07.721Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

Abstract

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
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.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Haukkanen, P., ‘Menon’s identity with respect to a generalized divisibility relation’, Aequationes Math. 70 (2005), 240–246.Google Scholar
Haukkanen, P. and McCarthy, P. J., ‘Sums of values of even functions’, Port. Math. 48 (1991), 53–66.Google Scholar
Haukkanen, P. and Sivaramakrishnan, R., ‘On certain trigonometric sums in several variables’, Collect. Math. 45 (1994), 245–261.Google Scholar
Haukkanen, P. and Wang, J., ‘A generalisation of Menon’s identity with respect to a set of polynomials’, Port. Math. 53 (1996), 331–337.Google Scholar
Li, Y. and Kim, D., ‘A Menon-type identity with many tuples of group of units in residually finite Dedekind domains’, J. Number Theory 175 (2017), 42–50.Google Scholar
Li, Y. and Kim, D., ‘Menon-type identities derived from actions of subgroups of general linear groups’, J. Number Theory 179 (2017), 97–112.Google Scholar
Li, Y., Hu, X. and Kim, D., ‘A Menon-type identity with multiplicative and additive characters’, Taiwanese J. Math. (2019), to appear.Google Scholar
McCarthy, P. J., Introduction to Arithmetical Functions, Universitext (Springer, New York, 1986).Google Scholar
Menon, P. K., ‘On the sum ∑(a - 1, n)[(a, n) = 1]’, J. Indian Math. Soc. 29 (1965), 155–163.Google Scholar
Miguel, C., ‘Menon’s identity in residually finite Dedekind domains’, J. Number Theory 137 (2014), 179–185.Google Scholar
Miguel, C., ‘A Menon-type identity in residually finite Dedekind domains’, J. Number Theory 164 (2016), 43–51.Google Scholar
Nageswara Rao, K., ‘On certain arithmetical sums’, in: The Theory of Arithmetic Functions, Lecture Notes in Mathematics, 251 (eds. Gioia, A. A. and Goldsmith, D. L.) (Springer, Berlin, Heidelberg, 1972), 181–192.Google Scholar
Neumann, P., ‘A lemma that is not Burnside’s’, Math. Sci. 4 (1979), 133–141.Google Scholar
Richards, I. M., ‘A remark on the number of cyclic subgroups of a finite group’, Amer. Math. Monthly 91 (1984), 571–572.Google Scholar
Schmidt, R., Subgroup Lattices of Groups, de Gruyter Expositions in Mathematics, 14 (de Gruyter, Berlin, 1994).Google Scholar
Sita Ramaiah, V., ‘Arithmetical sums in regular convolutions’, J. Reine Angew. Math. 303/304 (1978), 265–283.Google Scholar
Sury, B., ‘Some number-theoretic identities from group actions’, Rend. Circ. Mat. Palermo 58 (2009), 99–108.Google Scholar
Tărnăuceanu, M., ‘An arithmetic method of counting the subgroups of a finite abelian group’, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 53(101) (2010), 373–386.Google Scholar
Tărnăuceanu, M., ‘A generalization of Menon’s identity’, J. Number Theory 132 (2012), 2568–2573.Google Scholar
Tóth, L., ‘Menon’s identity and arithmetical sums representing functions of several variables’, Rend. Semin. Mat. Univ. Politec. Torino 69 (2011), 97–110.Google Scholar
Tóth, L., ‘On the number of cyclic subgroups of a finite abelian group’, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 55(103) (2012), 423–428.Google Scholar
Tóth, L., ‘Menon-type identities concerning Dirichlet characters’, Int. J. Number Theory 14 (2018), 1047–1054.Google Scholar
Zhao, X.-P. and Cao, Z.-F., ‘Another generalization of Menon’s identity’, Int. J. Number Theory 13 (2017), 2373–2379.Google Scholar