# Finite groups with the same conjugacy class sizes as a finite simple group

Document Type : Research Paper

Author

University of Shahre-kord

Abstract

For a finite group $H$‎, ‎let $cs(H)$ denote the set of non-trivial conjugacy class sizes of $H$ and $OC(H)$ be the set of the order components of $H$‎. ‎In this paper‎, ‎we show that if $S$ is a finite simple group with the disconnected prime graph and $G$ is a finite group such that $cs(S)=cs(G)$‎, ‎then $|S|=|G/Z(G)|$ and $OC(S)=OC(G/Z(G))$‎. ‎In particular‎, ‎we show that for some finite simple group $S$‎, ‎$G \cong S \times Z(G)$‎.

Keywords

Main Subjects

#### References

[1] R. Abb ott and et al., Atlas of nite group representations-version 3, brauer.maths.qmul.ac.uk/Atlas/v3/.
[2] N. Ahanjideh, Thompson's conjecture for nite simple groups of Lie typ e B n and Cn , J. Group Theory , 19
(2016) 713{733.
[3] N. Ahanjideh, Groups with the given set of the lengths of conjugacy classes, Turkish J. Math. , 39 (2015) 507{514.
[4] N. Ahanjideh, Thompson's conjecture for some nite simple groups with connected prime graph, Algebra Logic ,
51 (2013) 451{478.
[5] N. Ahanjideh, On the Thompson's conjecture on conjugacy classes sizes, Int. J. Algebr Comput. , 23 (2013)
37{68.
[6] N. Ahanjideh, On Thompson's conjecture for some nite simple groups, J. Algebra , 344 (2011) 205{228.
[7] N. Ahanjideh and M. Ahanjideh, On the validity of Thompson's conjecture for nite simple groups, Commun.
Algebra , 41 (2013) 4116{4145.
[8] Z. Akhlaghi, M. Khatami, T. Le, J. Mo ori and H. P. Tong-Viet, A dual version of Hupp ert's conjecture on
conjugacy class sizes, J. Group Theory , 18 (2015) 115{131.
[9] A. R. Camina and R. D. Camina, The in
uence of conjugacy class sizes on the structure of nite groups: a
survey, Asian-Eur. J. Math. , 4 (2011) 559{588.
[10] A. R. Camina and R. D. Camina, Recognising nilp otent groups, J. Algebra , 300 (2006) 16{24.
[11] G. Chen, Characterization of 3 D 4 ( q ), Southeast Asian Bul l. Math. , 25 (2001) 389{401.
[12] G. Chen, Characterization of Lie typ e group G 2 ( q ) by its order comp onents, J. Southwest China Normal Univ.
(Natural Science) , 26 (2001) 503{509.
[13] G. Chen, A new characterization of Suzuki-Ree groups, Sci. China Ser. A , 40 (1997) 807{812.
[14] G. Chen, A new characterization of sp oradic simple groups, Algebra Col loq. , 3 (1996) 49{58.
[15] G. Chen, A new characterization of G 2 ( q ), J. Southwest China Normal Univ. , 21 (1996) 47{51.
[16] G. Chen, A new characterization of E 8 ( q ), J. Southwest China Normal Univ. , 21 no. 3 (1996) 215{217.
[17] G. Chen, On Thompson's conjecture, J. Algebra , 185 (1996) 184{193.
[18] D. Chillag and M. Herzog, On the length of the conjugacy classes of nite groups, J. Algebra , 131 (1990) 110{125.
[19] M. R. Darafsheh, Characterization of the groups D p +1 (2) and D p +1 (3) using order comp onents, J. Korean Math.
Soc. , 47 (2010) 311{329.
[20] M. R. Darafsheh and A. Mahmiani, A characterization of the group 2 D n (2) where n = 2 m + 1  5, J. Appl.
Math. Comput. , 31 (2009) 447{457.
[21] A. Iranmanesh and B. Khosravi, A characterization of F4 ( q ) where q is an o dd prime p ower, Lecture Notes
London Math. Soc. , 304 (2003) 277{283.
[22] A. Iranmanesh and B. Khosravi, A characterization of C 2 ( q ), where q > 5, Commentationes Mathematicae
Universitatis Carolinae , 43 (2002) 9{21.
[23] A. Iranmanesh and B. Khosravi, A characterization of F4 ( q ) where q = 2 n ; ( n > 1), Far East J. Math. Sci. , 2
(2000) 853{859.
[24] B. Khosravi, B. Khosravi and B. Khosravi, A new characterization of P S U ( p; q ), Acta Math. Hungar. , 107
(2005) 235{252.
[25] A. Khosravi and B. Khosravi, r -Recognizability of B n ( q ) and C n ( q ) where n = 2 m  4, J. Pure Appl. Algebra ,
199 (2005) 149{165.
[26] A. Khosravi and B. Khosravi, A new characterization of P S L ( p; q ), Commun. Algebra , 32 (2004) 2325{2339.
[27] A. Khosravi and B. Khosravi, A characterization of 2 D n ( q ), where n = 2 m  4, Int. J. Math. Game Theory
Algebra , 13 (2003) 253{265.
[28] B. Khosravi and B. Khosravi, A characterization of 2 E 6 ( q ), Kumamoto J. Math. , 16 (2003) 1{11.
[29] B. Khosravi and B. Khosravi, A characterization of E 6 ( q ), Algebras, Groups and Geom. , 19 (2002) 225{243.
[30] E. I. Khukhro and V.D. Mazurov, Unsolved problems in group theory: The Kourovka Notebook , Sob olev Institute
of Mathematics, Novosibirsk, 17th edition, 2010.
[31] P. Kleidman and M. Lieb eck, The subgroup structure of nite classical groups , Cambridge University Press, 1990.
[32] A. S. Kondratiev, Prime graph comp onents of nite simple groups, Math. USSR-Sb. , 67 (1990) 235{247.
[33] G. Navarro, The set of conjugacy class sizes of a nite group do es not determine its solvability, J. Algebra , 411
(2014) 47{49.
[34] A. V. Vasilev, On Thompson's conjecture, Sib. Elektron. Mat. Izv. , 6 (2009) 457{464.
[35] J. S. Williams, Prime graph comp onents of nite groups, J. Algebra , 69 (1981) 487{513.