Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A characterization of Suzuki's simple groups


Author: Wu Jie Shi
Journal: Proc. Amer. Math. Soc. 114 (1992), 589-591
MSC: Primary 20D06
DOI: https://doi.org/10.1090/S0002-9939-1992-1074758-0
MathSciNet review: 1074758
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this short paper we have characterized Suzuki's simple groups $ {S_z}({2^{2m + 1}}),m \geqslant 1$ using only the set $ {\pi _e}(G)$ of orders of elements in the group $ G$. That is, we have

Theorem 2. Let $ G$ be a finite group. Then $ G \simeq {S_z}({2^{2m + 1}}),m \geqslant 1$ if and only if $ {\pi _e}(G) = \{ 2,4,all\;factors\;of\;{\text{(}}{{\text{2}}^{2m + 1}}{\text{ ... ...ext{2}}^{2m + 1}} - {2^{m + 1}} + 1),\;and\;({2^{2m + 1}} + {2^{m + 1}} + 1)\} $.


References [Enhancements On Off] (What's this?)

  • [1] R. Brandi, Finite groups all of whose elements are of prime power order, Boll. Un. Mat. Ital. A. (6) 18 (1981), 491-493. MR 633687 (82m:20012)
  • [2] G. Higman, Finite groups in which every element has prime power order, J. London Math. Soc. 32 (1957), 335-342. MR 0089205 (19:633d)
  • [3] B. Huppert and N. Blackburn, Finite Groups III, Springer-Verlag, Berlin, Heidelberg, and New York, 1982. MR 662826 (84i:20001b)
  • [4] Shi Wujie, A characteristic property of $ {J_1}$ and $ \operatorname{PSL}_2({2^n})$, Adv. in Math. 16 (1987), 397-401. (Chinese)
  • [5] -, A new characterization of some simple groups of Lie type, Contemp. Math. 82 (1989), 171-180. MR 982286 (90c:20022)
  • [6] -, A characteristic property of $ \operatorname{PSL}_2(7)$, J. Austral. Math. Soc. Ser. A 36 (1984), 354-356. MR 733907 (85h:20021)
  • [7] -, A characterization of some projective special linear groups, J. of Math. (PRC) 5 (1985), 191-200. MR 843760 (87h:20088)
  • [8] -, A characteristic property of $ {A_5}$, J. Southwest-China Teachers Univ. Ser. B no. 3 (1986), 11-14. (Chinese)
  • [9] M. Suzuki, A new type of simple groups of finite order, Proc. Nat. Acad. Sci. U. S. A. 46 (1960), 868-870. MR 0120283 (22:11038)
  • [10] -, Finite groups with nilpotent centralizers, Trans. Amer. Math. Soc. 99 (1961), 425-470. MR 0131459 (24:A1309)
  • [11] -, On a class of doubly transitive groups, Ann. Math. 75 (1962), 105-145. MR 0136646 (25:112)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 20D06

Retrieve articles in all journals with MSC: 20D06


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1992-1074758-0
Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society