Czechoslovak Mathematical Journal, first online, pp. 1-13

A variation of Thompson's conjecture for the symmetric groups

Mahdi Abedei, Ali Iranmanesh, Farrokh Shirjian

Received November 9, 2018.   Published online January 29, 2020.

Abstract:  Let $G$ be a finite group and let $N(G)$ denote the set of conjugacy class sizes of $G$. Thompson's conjecture states that if $G$ is a centerless group and $S$ is a non-abelian simple group satisfying $N(G)=N(S)$, then $G\cong S$. In this paper, we investigate a variation of this conjecture for some symmetric groups under a weaker assumption. In particular, it is shown that $G\cong{\rm Sym}(p+1)$ if and only if $|G|=(p+1)!$ and $G$ has a special conjugacy class of size $(p + 1)!/p$, where $p>5$ is a prime number. Consequently, if $G$ is a centerless group with $N(G)=N({\rm Sym}(p+1))$, then $G \cong{\rm Sym}(p+1)$.
Keywords:  Thompson's conjecture; conjugacy class size; symmetric groups; prime graph
Classification MSC:  20D08, 20D60
DOI:  10.21136/CMJ.2020.0501-18

PDF available at:  Springer   Institute of Mathematics CAS

[1] N. Ahanjideh: On Thompson's conjecture for some finite simple groups. J. Algebra 344 (2011), 205-228. DOI 10.1016/j.jalgebra.2011.05.043 | MR 2831937 | Zbl 1247.20015
[2] A. K. Asboei, M. R. Darafsheh, R. Mohammadyari: The influence of order and conjugacy class length on the structure of finite groups. Hokkaido Math. J. 47 (2018), 25-32. DOI 10.14492/hokmj/1520928059 | MR 3773724 | Zbl 06853590
[3] A. K. Asboei, R. Mohammadyari: Characterization of the alternating groups by their order and one conjugacy class length. Czech. Math. J. 66 (2016), 63-70. DOI 10.1007/s10587-016-0239-0 | MR 3483222 | Zbl 1374.20008
[4] A. K. Asboei, R. Mohammadyari: Recognizing alternating groups by their order and one conjugacy class length. J. Algebra Appl. 15 (2016), Article ID 1650021. DOI 10.1142/S0219498816500213 | MR 3405720 | Zbl 1336.20026
[5] A. K. Asboei, R. Mohammadyari: New characterization of symmetric groups of prime degree. Acta Univ. Sapientiae, Math. 9 (2017), 5-12. DOI 10.1515/ausm-2017-0001 | MR 3684822 | Zbl 1370.20013
[6] G. Chen: On Thompson's conjecture for sporadic groups. Proc. of the First Academic Annual Meeting of Youth Fujian Science and Technology Publishing House, Fuzhou (1992), 1-6. (In Chinese.) MR 1252902
[7] G. Chen: On Thompson's Conjecture, PhD Thesis. Sichuan University, Chengdu (1994).
[8] G. Chen: On the structure of Frobenius group and 2-Frobenius group. J. Southwest China Normal. Univ. 20 (1995), 485-487. (In Chinese.)
[9] G. Chen: On Thompson's conjecture. J. Algebra 185 (1996), 184-193. DOI 10.1006/jabr.1996.0320 | MR 1409982 | Zbl 0861.20018
[10] G. Chen: Further reflections on Thompson's conjecture. J. Algebra 218 (1999), 276-285. DOI 10.1006/jabr.1998.7839 | MR 1704687 | Zbl 0931.20020
[11] Y. Chen, G. Chen, J. Li: Recognizing simple $K_4$-groups by few special conjugacy class sizes. Bull. Malays. Math. Sci. Soc. (2) 38 (2015), 51-72. DOI 10.1007/s40840-014-0003-2 | MR 3394038 | Zbl 1406.20016
[12] D. Gorenstein: Finite Groups. Chelsea Publishing, New York (1980). MR 0569209 | Zbl 0463.20012
[13] A. Iranmanesh, S. H. Alavi, B. Khosravi: A characterization of $PSL(3,q)$ where $q$ is an odd prime power. J. Pure Appl. Algebra 170 (2002), 243-254. DOI 10.1016/S0022-4049(01)00113-X | MR 1904845 | Zbl 1001.20005
[14] A. S. Kondrat'ev, V. D. Mazurov: Recognition of alternating groups of prime degree from their element orders. Sib. Math. J. 41 (2000), 294-302 (In English. Russian original.); translation from Sib. Mat. Zh. 41 (2000), 359-369. DOI 10.1007/BF02674599 | MR 1762188 | Zbl 0956.20007
[15] J. B. Li: Finite Groups with Special Conjugacy Class Sizes or Generalized Permutable Subgroups, Ph.D. Thesis. Southwest University, Chongqing (2012).
[16] V. D. Mazurov, E. I. Khukhro (eds.): The Kourovka Notebook: Unsolved Problems in Group Theory. Institute of Mathematics, Russian Academy of Sciences, Siberian Div., Novosibirsk (2018). MR 3408705 | Zbl 1372.20001
[17] W. J. Shi, J. X. Bi: A new characterization of the alternating groups. Southeast Asian Bull. Math. 16 (1992), 81-90. MR 1173612 | Zbl 0790.20030
[18] A. V. Vasil'ev: On Thompson's conjecture. Sib. Elektron. Mat. Izv. 6 (2009), 457-464. MR 2586699 | Zbl 1289.20057
[19] J. S. Williams: Prime graph components of finite groups. J. Algebra 69 (1981), 487-513. DOI 10.1016/0021-8693(81)90218-0 | MR 0617092 | Zbl 0471.20013

Affiliations:   Mahdi Abedei, Ali Iranmanesh (corresponding author), Farrokh Shirjian, Department of Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, P.O. Box 14115-137, Tehran, Iran, e-mail:,,

PDF available at: