Open Access

Subgroups of Finite Abelian Groups Having Rank Two via Goursat’s Lemma

   | Mar 11, 2015

Cite

[1] ANDERSON, D. D.-CAMILLO, V.: Subgroups of direct products of groups, ideals and subrings of direct products of rings, and Goursat’s lemma, in: Rings, Modules and Representations. Internat. Conf. on Rings and Things, Zanesville, OH, USA, 2007, Contemp. Math., Vol. 480, Amer. Math. Soc., Providence, RI, 2009, pp. 1-12.10.1090/conm/480/09364Search in Google Scholar

[2] BAUER, K.-SEN, D.-ZVENGROWSKI, P.: A generalized Goursat lemma, Preprint, 2011, arXiv: 11009.0024 [math.GR].Search in Google Scholar

[3] CALHOUN, W. C.: Counting the subgroups of some finite groups, Amer. Math. Monthly 94 (1987), 54-59.10.1080/00029890.1987.12000593Search in Google Scholar

[4] CĂLUGĂREANU, G.: The total number of subgroups of a finite abelian group, Sci. Math. Jpn. 60 (2004), 157-167.Search in Google Scholar

[5] CRAWFORD, R. R.-WALLACE, K. D.: On the number of subgroups of index two- -An application of Goursat’s theorem for groups, Math. Mag. 48 (1975), 172-174.Search in Google Scholar

[6] GOURSAT, É.: Sur les substitutions orthogonales et les divisions régulières de l’espace, Ann. Sci. Ècole Norm. Sup. (3) 6 (1889), 9-102.10.24033/asens.317Search in Google Scholar

[7] HAMPEJS, M.-HOLIGHAUS, N.-TÓTH, L.-WIESMEYR, C.: Representing and counting the subgroups of the group Zm × Zn, Journal of Numbers, Vol. 2014, Article ID 491428.10.1155/2014/491428Search in Google Scholar

[8] HAMPEJS, M.-TÓTH, L.: On the subgroups of finite abelian groups of rank three, Ann. Univ. Sci. Budapest. Eötvös Sect. Comput. 39 (2013), 111-124.Search in Google Scholar

[9] LAMBEK, J.: Goursat’s theorem and the Zassenhaus lemma, Canad. J. Math. 10 (1958), 45-56.10.4153/CJM-1958-005-6Search in Google Scholar

[10] MACHÌ, A.: Groups. An Introduction to Ideas and Methods of the Theory of Groups. Springer, Berlin, 2012.10.1007/978-88-470-2421-2Search in Google Scholar

[11] NOWAK, W. G.-TÓTH, L.: On the average number of subgroups of the group Zm×Zn, Int. J. Number Theory 10 (2014), 363-374.10.1142/S179304211350098XSearch in Google Scholar

[12] PETRILLO, J.: Goursat’s other theorem, College Math. J. 40 (2009), 119-124.10.4169/193113409X469569Search in Google Scholar

[13] PETRILLO, J.: Counting subgroups in a direct product of finite cyclic groups, College Math. J. 42 (2011), 215-222.10.4169/college.math.j.42.3.215Search in Google Scholar

[14] ROTMAN, J. J.: An Introduction to the Theory of Groups (4th ed.), in: Grad. Texts in Math., Vol. 148, Springer-Verlag, New York, 1995.Search in Google Scholar

[15] 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.Search in Google Scholar

[16] TÓTH, L.: Menon’s identity and arithmetical sums representing functions of several variables, Rend. Sem. Mat. Univ. Politec. Torino 69 (2011), 97-110.Search in Google Scholar

[17] 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.Search in Google Scholar

[18] TÓTH, L.: Multiplicative arithmetic functions of several variables: a survey, in: Mathematics Without Boundaries, Surveys in Pure Mathematics (Th. M. Rassias, P. Pardalos, eds.), Springer, New York, 2014, pp. 483-514.10.1007/978-1-4939-1106-6_19Search in Google Scholar

eISSN:
1210-3195
Language:
English
Publication timeframe:
3 times per year
Journal Subjects:
Mathematics, General Mathematics