Previous |  Up |  Next

Article

Full entry | Fulltext not available (moving wall 24 months)      Feedback
Keywords:
Thompson's conjecture; conjugacy class size; symmetric groups; prime graph
Summary:
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 $G \cong {\rm Sym}(p+1)$.
References:
[1] Ahanjideh, N.: 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] Asboei, A. K., Darafsheh, M. R., Mohammadyari, R.: 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] Asboei, A. K., Mohammadyari, R.: 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] Asboei, A. K., Mohammadyari, R.: 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] Asboei, A. K., Mohammadyari, R.: 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] Chen, G.: 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 Chinese. MR 1252902
[7] Chen, G.: On Thompson's Conjecture, PhD Thesis. Sichuan University, Chengdu (1994). MR 1409982
[8] Chen, G.: On the structure of Frobenius group and 2-Frobenius group. J. Southwest China Normal. Univ. 20 (1995), 485-487 Chinese.
[9] Chen, G.: On Thompson's conjecture. J. Algebra 185 (1996), 184-193. DOI 10.1006/jabr.1996.0320 | MR 1409982 | Zbl 0861.20018
[10] Chen, G.: Further reflections on Thompson's conjecture. J. Algebra 218 (1999), 276-285. DOI 10.1006/jabr.1998.7839 | MR 1704687 | Zbl 0931.20020
[11] Chen, Y., Chen, G., Li, J.: 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] Gorenstein, D.: Finite Groups. Chelsea Publishing, New York (1980). MR 0569209 | Zbl 0463.20012
[13] Iranmanesh, A., Alavi, S. H., Khosravi, B.: 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] Kondrat'ev, A. S., Mazurov, V. D.: Recognition of alternating groups of prime degree from their element orders. Sib. Math. J. 41 (2000), 294-302 English. Russian original translation from Sib. Mat. Zh. 41 2000 359-369. DOI 10.1007/BF02674599 | MR 1762188 | Zbl 0956.20007
[15] Li, J. B.: Finite Groups with Special Conjugacy Class Sizes or Generalized Permutable Subgroups, Ph.D. Thesis. Southwest University, Chongqing (2012).
[16] Mazurov, V. D., (eds.), E. I. Khukhro: 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] Shi, W. J., Bi, J. X.: A new characterization of the alternating groups. Southeast Asian Bull. Math. 16 (1992), 81-90. MR 1173612 | Zbl 0790.20030
[18] Vasil'ev, A. V.: On Thompson's conjecture. Sib. Elektron. Mat. Izv. 6 (2009), 457-464. MR 2586699 | Zbl 1289.20057
[19] Williams, J. S.: 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
Partner of
EuDML logo