Title:
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A variation of Thompson's conjecture for the symmetric groups (English) |
Author:
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Abedei, Mahdi |
Author:
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Iranmanesh, Ali |
Author:
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Shirjian, Farrokh |
Language:
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English |
Journal:
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Czechoslovak Mathematical Journal |
ISSN:
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0011-4642 (print) |
ISSN:
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1572-9141 (online) |
Volume:
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70 |
Issue:
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3 |
Year:
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2020 |
Pages:
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743-755 |
Summary lang:
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English |
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Category:
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math |
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Summary:
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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 $G \cong {\rm Sym}(p+1)$. (English) |
Keyword:
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Thompson's conjecture |
Keyword:
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conjugacy class size |
Keyword:
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symmetric groups |
Keyword:
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prime graph |
MSC:
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20D08 |
MSC:
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20D60 |
idZBL:
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07250686 |
idMR:
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MR4151702 |
DOI:
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10.21136/CMJ.2020.0501-18 |
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Date available:
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2020-09-07T09:37:30Z |
Last updated:
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2022-10-03 |
Stable URL:
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http://hdl.handle.net/10338.dmlcz/148325 |
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Reference:
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[1] Ahanjideh, N.: On Thompson's conjecture for some finite simple groups.J. Algebra 344 (2011), 205-228. Zbl 1247.20015, MR 2831937, 10.1016/j.jalgebra.2011.05.043 |
Reference:
|
[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. Zbl 06853590, MR 3773724, 10.14492/hokmj/1520928059 |
Reference:
|
[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. Zbl 1374.20008, MR 3483222, 10.1007/s10587-016-0239-0 |
Reference:
|
[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. Zbl 1336.20026, MR 3405720, 10.1142/S0219498816500213 |
Reference:
|
[5] Asboei, A. K., Mohammadyari, R.: New characterization of symmetric groups of prime degree.Acta Univ. Sapientiae, Math. 9 (2017), 5-12. Zbl 1370.20013, MR 3684822, 10.1515/ausm-2017-0001 |
Reference:
|
[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 |
Reference:
|
[7] Chen, G.: On Thompson's Conjecture, PhD Thesis.Sichuan University, Chengdu (1994). MR 1409982 |
Reference:
|
[8] Chen, G.: On the structure of Frobenius group and 2-Frobenius group.J. Southwest China Normal. Univ. 20 (1995), 485-487 Chinese. |
Reference:
|
[9] Chen, G.: On Thompson's conjecture.J. Algebra 185 (1996), 184-193. Zbl 0861.20018, MR 1409982, 10.1006/jabr.1996.0320 |
Reference:
|
[10] Chen, G.: Further reflections on Thompson's conjecture.J. Algebra 218 (1999), 276-285. Zbl 0931.20020, MR 1704687, 10.1006/jabr.1998.7839 |
Reference:
|
[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. Zbl 1406.20016, MR 3394038, 10.1007/s40840-014-0003-2 |
Reference:
|
[12] Gorenstein, D.: Finite Groups.Chelsea Publishing, New York (1980). Zbl 0463.20012, MR 0569209 |
Reference:
|
[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. Zbl 1001.20005, MR 1904845, 10.1016/S0022-4049(01)00113-X |
Reference:
|
[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. Zbl 0956.20007, MR 1762188, 10.1007/BF02674599 |
Reference:
|
[15] Li, J. B.: Finite Groups with Special Conjugacy Class Sizes or Generalized Permutable Subgroups, Ph.D. Thesis.Southwest University, Chongqing (2012). |
Reference:
|
[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). Zbl 1372.20001, MR 3408705 |
Reference:
|
[17] Shi, W. J., Bi, J. X.: A new characterization of the alternating groups.Southeast Asian Bull. Math. 16 (1992), 81-90. Zbl 0790.20030, MR 1173612 |
Reference:
|
[18] Vasil'ev, A. V.: On Thompson's conjecture.Sib. Elektron. Mat. Izv. 6 (2009), 457-464. Zbl 1289.20057, MR 2586699 |
Reference:
|
[19] Williams, J. S.: Prime graph components of finite groups.J. Algebra 69 (1981), 487-513. Zbl 0471.20013, MR 0617092, 10.1016/0021-8693(81)90218-0 |
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