# A CHARACTERIZATION OF SOME PGL(2, q) BY MAXIMUM ELEMENT ORDERS

• LI, JINBAO (Chongqing Key Laboratory of Group & Graph Theories and Applications Chongqing University of Arts and Sciences) ;
• SHI, WUJIE (Chongqing Key Laboratory of Group & Graph Theories and Applications Chongqing University of Arts and Sciences) ;
• YU, DAPENG (Chongqing Key Laboratory of Group & Graph Theories and Applications Chongqing University of Arts and Sciences)
• Received : 2014.10.16
• Published : 2015.11.30
• 72 8

#### Abstract

In this paper, we characterize some PGL(2, q) by their orders and maximum element orders. We also prove that PSL(2, p) with $p{\geqslant}3$ a prime can be determined by their orders and maximum element orders. Moreover, we show that, in general, if $q=p^n$ with p a prime and n > 1, PGL(2, q) can not be uniquely determined by their orders and maximum element orders. Several known results are generalized.

#### Keywords

finite simple groups;maximum element order;characterization

#### References

1. B. Huppert, Endliche Gruppen I, Springer-Verlag, Heidelberg-New York, 1967.
2. W. M. Kantor, Linear groups containing a Singer cycle, J. Algebra 62 (1980), no. 1, 232-234. https://doi.org/10.1016/0021-8693(80)90214-8
3. W. M. Kantor and A. Seress, Large element orders and the characteristic of Lie-type simple groups, J. Algebra 322 (2009), no. 3, 802-832. https://doi.org/10.1016/j.jalgebra.2009.05.004
4. W. J. Shi, A new characterization of the sporadic simple groups, Group Theory: Proc. of the 1987 Singapore Conf., 531-540, Walter de Gruyter, Berlin-New York, 1989.
5. W. J. Shi, A new characterization of some simple groups of Lie type, Classical groups and related topics (Beijing, 1987), 171-180, Contemp. Math., 82, Amer. Math. Soc., Providence, RI, 1989.
6. W. J. Shi, On a problem of E. Artin, Acta Math. Sinica 35 (1992), no. 2, 262-265.
7. W. J. Shi, The pure quantitative characterization of finite groups (I), Progr. Natur. Sci. 4 (1994), 316-326.
8. W. J. Shi, Pure quantitative characterization of finite simple groups, Front. Math. China 2 (2007), no. 1, 123-125. https://doi.org/10.1007/s11464-007-0008-3
9. W. J. Shi and J. X. Bi, A characteristic property for each finite projective special linear groups, GroupsCanberra 1989, 171-180, Lecture Notes in Math., 1456, Springer, Berlin, 1990.
10. W. J. Shi, A characterization of Suzuki-Ree groups, Sci. China Ser. A 34 (1991), no. 1, 14-19.
11. W. J. Shi, A characterization of the alternating groups, SEA Bull. Math. 16 (1992), no. 1, 81-90.
12. A. V. Vasil'ev, M. A. Grechkoseeva, and V. D. Mazurov, Characterization of the finite simple groups by spectra and order, Algebra and Logic 48 (2009), 385-409. https://doi.org/10.1007/s10469-009-9074-9
13. J. S. Williams, Prime graph components of finite groups, J. Algebra 69 (1981), no. 2, 487-513. https://doi.org/10.1016/0021-8693(81)90218-0
14. L. G. He and G. Y. Chen, A new characterization of simple $K_3$-groups, Comm. Algebra 40 (2012), no. 10, 3903-3911. https://doi.org/10.1080/00927872.2011.598595
15. L. G. He and G. Y. Chen, A new characterization of simple $K_4$-groups with type $L_2$(p), Adv. Math. (China) 43 (2014), no. 5, 667-670.
16. M. C. Xu and W. J. Shi, Pure quantitative characterization of finite simple groups $^{2}D_n(q)$ and $D_l(q)$ (l odd), Algebra Colloq. 10 (2003), no. 3, 427-443.
17. Q. L. Zhang and W. J. Shi, A new characterization of simple $K_3$-groups and some PSL(2, p), Algebra Colloq. 20 (2013), no. 3, 361-368. https://doi.org/10.1142/S1005386713000333
18. R. Brauer and W. F. Reynolds, On a problem of E. Artin, Ann. of Math. 68 (1958), no. 3, 713-720. https://doi.org/10.2307/1970164
19. H. P. Cao and W. J. Shi, Pure quantitative characterization of finite projective special unitary groups, Sci, China Ser. A 45 (2002), no. 6, 761-772.
20. G. Y. Chen, On Thompson's cconjecture for sporadic simple groups, Proc. China Assoc. Sci. and Tech. First Academic Annual Meeting of Youths, Chinese Sci. and Tech., 1-6, Press, Beijing, 1992.
21. G. Y. Chen, A new characterization of sporadic simple groups, Algebra Colloq. 3 (1996), no. 1, 49-58.
22. G. Y. Chen, On Thompson's conjecture, J. Algebra 185 (1996), no. 1, 184-193. https://doi.org/10.1006/jabr.1996.0320
23. G. Y. Chen, Further reflections on Thompson's conjecture, J. Algebra 218 (1999), no. 1, 276-285. https://doi.org/10.1006/jabr.1998.7839
24. G. Y. Chen, V. D. Mazurov, W. J. Shi, A. V. Vasil'ev, and A. K. Zhurtov, Recognition of finite almost simple groups PGL(2, q) by their spectrum, J. Group Theory 10 (2007), no. 1, 71-85.
25. J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, Atlas of Finite Groups, Clarendon Press, Oxford, 1985.
26. L. E. Dickson, Linear Groups, Dover Publications, New York, 1958.
27. M. A. Grechkoseeva, On difference between the spectra of the simple groups $B_n(q)$$and$C_n(q)\$, Siberian Math. J. 48 (2007), 73-75. https://doi.org/10.1007/s11202-007-0008-2
28. S. Guest, J. Morris, C. E. Praeger, and P. Spiga, On the maximum orders of elements of finite almost simple groups and primitive permutation groups, ArXiv: 1301.5166v1.
29. W. B. Guo, The Theory of Classes of Groups, Science Press-Kluwer Academic Publishers, Beijing-New York-Dorlrecht-Boston-London, 2000.