Czechoslovak Mathematical Journal, Vol. 73, No. 1, pp. 311-317, 2023


On a group-theoretical generalization of the Gauss formula

Georgiana Fasolă, Marius Tărnăuceanu

Received May 28, 2022.   Published online December 15, 2022.

Abstract:  We discuss a group-theoretical generalization of the well-known Gauss formula involving the function that counts the number of automorphisms of a finite group. This gives several characterizations of finite cyclic groups.
Keywords:  Gauss formula; Euler's totient function; automorphism group; finite group; cyclic group; abelian group
Classification MSC:  20D60, 11A25, 20D99, 11A99


References:
[1] S. J. Baishya: Revisiting the Leinster groups. C. R., Math., Acad. Sci. Paris 352 (2014), 1-6. DOI 10.1016/j.crma.2013.11.009 | MR 3150758 | Zbl 1292.20026
[2] S. J. Baishya, A. K. Das: Harmonic numbers and finite groups. Rend. Semin. Mat. Univ. Padova 132 (2014), 33-43. DOI 10.4171/RSMUP/132-3 | MR 3276824 | Zbl 1314.20020
[3] J. N. S. Bidwell, M. J. Curran, D. J. McCaughan: Automorphisms of direct products of finite groups. Arch. Math. 86 (2006), 481-489. DOI 10.1007/s00013-005-1547-z | MR 2241597 | Zbl 1103.20016
[4] J. N. Bray, R. A. Wilson: On the orders of automorphism groups of finite groups. Bull. Lond. Math. Soc. 37 (2005), 381-385. DOI 10.1112/S002460930400400X | MR 2131391 | Zbl 1072.20029
[5] J. N. Bray, R. A. Wilson: On the orders of automorphism groups of finite groups II. J. Group Theory 9 (2006), 537-547. DOI 10.1515/JGT.2006.036 | MR 2243245 | Zbl 1103.20017
[6] T. De Medts, A. Maróti: Perfect numbers and finite groups. Rend. Semin. Mat. Univ. Padova 129 (2013), 17-33. DOI 10.4171/RSMUP/129-2 | MR 3090628 | Zbl 1280.20026
[7] T. De Medts, M. Tărnăuceanu: Finite groups determined by an inequality of the orders of their subgroups. Bull. Belg. Math. Soc. - Simon Stevin 15 (2008), 699-704. DOI 10.36045/bbms/1225893949 | MR 2475493 | Zbl 1166.20017
[8] J. González-Sánchez, A. Jaikin-Zapirain: Finite $p$-groups with small automorphism group. Forum Math. Sigma 3 (2015), Article ID e7, 11 pages. DOI 10.1017/fms.2015.6 | MR 3376735 | Zbl 1319.20019
[9] C. J. Hillar, D. L. Rhea: Automorphisms of finite abelian groups. Am. Math. Mon. 114 (2007), 917-923. DOI 10.1080/00029890.2007.11920485 | MR 2363058 | Zbl 1156.20046
[10] I. M. Isaacs: Finite Group Theory. Graduate Studies in Mathematics 92. AMS, Providence (2008). DOI 10.1090/gsm/092 | MR 2426855 | Zbl 1169.20001
[11] G. A. Miller, H. C. Moreno: Non-abelian groups in which every subgroup is abelian. Trans. Am. Math. Soc. 4 (1903), 398-404. DOI 10.2307/1986409 | MR 1500650 | JFM 34.0173.01
[12] A. Sehgal, S. Sehgal, P. K. Sharma: The number of automorphism of a finite abelian group of rank two. J. Discrete Math. Sci. Cryptography 19 (2016), 163-171. DOI 10.1080/09720529.2015.1103469 | MR 3502205 | Zbl 07509102
[13] M. Tărnăuceanu: A generalization of the Euler's totient function. Asian-Eur. J. Math. 8 (2015), Article ID 1550087, 13 pages. DOI 10.1142/S1793557115500874 | MR 3424162 | Zbl 1336.20029
[14] M. Tărnăuceanu: Finite groups determined by an inequality of the orders of their subgroups II. Commun. Algebra 45 (2017), 4865-4868. DOI 10.1080/00927872.2017.1284228 | MR 3670357 | Zbl 1375.20025

Affiliations:   Georgiana Fasolă, Marius Tărnăuceanu (corresponding author), Faculty of Mathematics, Alexandru Ioan Cuza University, Bulevardul Carol I no. 11, 700506 Iaşi, Romania, e-mail: georgiana.fasola@student.uaic.ro, tarnauc@uaic.ro


 
PDF available at: