Czechoslovak Mathematical Journal, Vol. 67, No. 4, pp. 1049-1058, 2017


Thompson's conjecture for the alternating group of degree $2p$ and $2p+1$

Azam Babai, Ali Mahmoudifar

Received July 25, 2016.   First published October 5, 2017.

Abstract:  For a finite group $G$ denote by $N(G)$ the set of conjugacy class sizes of $G$. In 1980s, J. G. Thompson posed the following conjecture: If $L$ is a finite nonabelian simple group, $G$ is a finite group with trivial center and $N(G) = N(L)$, then $G\cong L$. We prove this conjecture for an infinite class of simple groups. Let $p$ be an odd prime. We show that every finite group $G$ with the property $Z(G)=1$ and $N(G) = N(A_i)$ is necessarily isomorphic to $A_i$, where $i\in\{2p,2p+1\}$.
Keywords:  finite group; conjugacy class size; simple group
Classification MSC:  20D05, 20D60


References:
[1] A. Abdollahi, H. Shahverdi: Characterization of the alternating group by its non-commuting graph. J. Algebra 357 (2012), 203-207. DOI 10.1016/j.jalgebra.2012.01.038 | MR 2905249 | Zbl 1255.20026
[2] 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
[3] N. Ahanjideh: On the Thompson's conjecture on conjugacy classes sizes. Int. J. Algebra Comput. 23 (2013), 37-68. DOI 10.1142/S0218196712500774 | MR 3040801 | Zbl 1281.20015
[4] S. H. Alavi, A. Daneshkhah: A new characterization of alternating and symmetric groups. J. Appl. Math. Comput. 17 (2005), 245-258. DOI 10.1007/BF02936052 | MR 2108803 | Zbl 1066.20012
[5] G. Chen: On Thompson's conjecture. J. Algebra 185 (1996), 184-193. DOI 10.1006/jabr.1996.0320 | MR 1409982 | Zbl 0861.20018
[6] I. B. Gorshkov: Thompson's conjecture for simple groups with connected prime graph. Algebra Logic 51 (2012), 111-127; translated from Algebra Logika 51 (2012), 168-192. (In Russian.) DOI 10.1007/s10469-012-9175-8 | MR 2986578 | Zbl 1270.20010
[7] I. B. Gorshkov: On Thompson's conjecture for alternating and symmetric groups of degree greater than 1361. Proc. Steklov Inst. Math. 293 (2016), S58-S65; translated from Tr. Inst. Mat. Mekh. (Ekaterinburg) 22 (2016), 44-51. (In Russian.) DOI 10.1134/S0081543816050060 | MR 3497182 | Zbl 1352.20022
[8] I. B. Gorshkov: Towards Thompson's conjecture for alternating and symmetric groups. J. Group Theory 19 (2016), 331-336. DOI 10.1515/jgth-2015-0043 | MR 3466599 | Zbl 1341.20022
[9] I. M. Isaacs: Finite Group Theory. Graduate Studies in Mathematics 92, American Mathematical Society, Providence (2008). DOI 10.1090/gsm/092 | MR 2426855 | Zbl 1169.20001
[10] A. Mahmoudifar, B. Khosravi: On the characterizability of alternating groups by order and prime graph. Sib. Math. J. 56 (2015), 125-131; translated from Sib. Mat. Zh. 56 (2015), 149-157. (In Russian.) DOI 10.1134/S0037446615010127 | MR 3407946 | Zbl 1318.20027
[11] V. D. Mazurov, E. I. Khukhro eds.: The Kourovka Notebook. Unsolved Problems in Group Theory. Institute of Mathematics, Russian Academy of Sciences Siberian Division, Novosibirsk (2010). MR 3235009 | Zbl 1211.20001
[12] I. A. Vakula: On the structure of finite groups isospectral to an alternating group. Proc. Steklov Inst. Math. 272 (2011), 271-286; translated from Tr. Inst. Mat. Mekh. (Ekaterinburg) 16 (2010), 45-60. (In Russian.) DOI 10.1134/S0081543811020192 | MR 3546195 | Zbl 1233.20016
[13] A. V. Vasil'ev: On Thompson's conjecture. Sib. Elektron. Mat. Izv. 6 (2009), 457-464. MR 2586699 | Zbl 1289.20057
[14] M. Xu: Thompson's conjecture for alternating group of degree 22. Front. Math. China 8 (2013), 1227-1236. DOI 10.1007/s11464-013-0320-z | MR 3091135 | Zbl 1281.20018

Affiliations:   Azam Babai, Department of Mathematics, University of Qom, Alghadir Blvd., 37185-3766 Qom, Iran, e-mail: a_babai@aut.ac.ir; Ali Mahmoudifar, Department of Mathematics, Tehran-North Branch, Islamic Azad University, South Makran Street, 1651153311 Tehran, Iran, e-mail: a_mahmoodifar@iau-tnb.ac.ir


 
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