The order of a group of even order

Author:
Hiroyoshi Yamaki

Journal:
Proc. Amer. Math. Soc. **136** (2008), 397-402

MSC (2000):
Primary 20D05, 20D06

Published electronically:
October 25, 2007

MathSciNet review:
2358476

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We will give an estimation of the order of a group of even order by the order of the centralizer of an involution using the classification of finite simple groups.

**1.**Michael Aschbacher and Gary M. Seitz,*Involutions in Chevalley groups over fields of even order*, Nagoya Math. J.**63**(1976), 1–91. MR**0422401****2.**Richard Brauer and K. A. Fowler,*On groups of even order*, Ann. of Math. (2)**62**(1955), 565–583. MR**0074414****3.**N. Burgoyne and C. Williamson,*Centralizers of involutions in Chevalley groups of odd characteristic*, Mimeographed notes (1972).**4.**Naoki Chigira, Nobuo Iiyori, and Hiroyoshi Yamaki,*Non-abelian Sylow subgroups of finite groups of even order*, Invent. Math.**139**(2000), no. 3, 525–539. MR**1738059**, 10.1007/s002229900040**5.**J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson,*Atlas of finite groups*, Oxford University Press, Eynsham, 1985. Maximal subgroups and ordinary characters for simple groups; With computational assistance from J. G. Thackray. MR**827219****6.**R. H. Dye,*On the conjugacy classes of involutions of the simple orthogonal groups over perfect fields of characteristic two*, J. Algebra**18**(1971), 414–425. MR**0276366****7.**R. H. Dye,*On the involution classes of the linear groups 𝐺𝐿_{𝑛}(𝐾), 𝑆𝐿_{𝑛}(𝐾), 𝑃𝐺𝐿_{𝑛}(𝐾), 𝑃𝑆𝐿_{𝑛}(𝐾) over fields of characteristic two*, Proc. Cambridge Philos. Soc.**72**(1972), 1–6. MR**0294519****8.**R. H. Dye,*On the conjugacy classes of involutions of the unitary groups 𝑈_{𝑚}(𝐾), 𝑆𝑈_{𝑚}(𝐾), 𝑃𝑈_{𝑚}(𝐾), 𝑃𝑆𝑈_{𝑚}(𝐾), over perfect fields of characteristic 2*, J. Algebra**24**(1973), 453–459. MR**0308287****9.**K. Harada and M. Miyamoto,*On the order of a group of even order*, To appear in J. Algebra.**10.**Bertram Huppert and Norman Blackburn,*Finite groups. III*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 243, Springer-Verlag, Berlin-New York, 1982. MR**662826****11.**Nobuo Iiyori and Hiroyoshi Yamaki,*Prime graph components of the simple groups of Lie type over the field of even characteristic*, J. Algebra**155**(1993), no. 2, 335–343. MR**1212233**, 10.1006/jabr.1993.1048**12.**A. S. Kondrat′ev,*On prime graph components of finite simple groups*, Mat. Sb.**180**(1989), no. 6, 787–797, 864 (Russian); English transl., Math. USSR-Sb.**67**(1990), no. 1, 235–247. MR**1015040****13.**Michio Suzuki,*Group theory. II*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 248, Springer-Verlag, New York, 1986. Translated from the Japanese. MR**815926****14.**J. S. Williams,*Prime graph components of finite groups*, J. Algebra**69**(1981), no. 2, 487–513. MR**617092**, 10.1016/0021-8693(81)90218-0

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
20D05,
20D06

Retrieve articles in all journals with MSC (2000): 20D05, 20D06

Additional Information

**Hiroyoshi Yamaki**

Affiliation:
Department of Mathematics, Kumamoto University, Kumamoto 860-8555 Japan

Address at time of publication:
JICA, Maipu 1300, Piso 21, C1006ACT Buenos Aires, Argentina

Email:
yamaki@gpo.kumamoto-u.ac.jp, yamaki.hiroyoshi@gmail.com

DOI:
https://doi.org/10.1090/S0002-9939-07-09118-6

Keywords:
Finite simple groups,
centralizers of involutions

Received by editor(s):
August 15, 2006

Published electronically:
October 25, 2007

Additional Notes:
The author was supported in part by Grant-in-Aid for Scientific Research (No. 16540030), Japan Society for the Promotion of Science

Communicated by:
Jonathan I. Hall

Article copyright:
© Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.