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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
Posted:
October 25, 2007
MathSciNet review:
2358476
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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
(54 #10391)
- 2.
Richard
Brauer and K.
A. Fowler, On groups of even order, Ann. of Math. (2)
62 (1955), 565–583. MR 0074414
(17,580e)
- 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
(2001c:20042), http://dx.doi.org/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
(88g:20025)
- 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
(43 #2113)
- 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
(45 #3589)
- 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
(46 #7401)
- 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, 1982. MR 662826
(84i:20001b)
- 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
(94e:05268), http://dx.doi.org/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
(90h:20018)
- 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 (87e:20001)
- 14.
J.
S. Williams, Prime graph components of finite groups, J.
Algebra 69 (1981), no. 2, 487–513. MR 617092
(82j:20054), http://dx.doi.org/10.1016/0021-8693(81)90218-0
- 1.
- M. Aschbacher and G. Seitz, Involutions in Chevalley groups over fields of even order, Nagoya Math. J. 63 (1976), 1-91. MR 0422401 (54:10391)
- 2.
- R. Brauer and K. A. Fowler, On groups of even order, Ann. Math. 62 (1955), 565-583. MR 0074414 (17:580e)
- 3.
- N. Burgoyne and C. Williamson, Centralizers of involutions in Chevalley groups of odd characteristic, Mimeographed notes (1972).
- 4.
- N. Chigira, N. Iiyori and H. Yamaki, Non-abelian Sylow subgroups of finite groups of even order, Invent. Math. 139 (2000), 525-539. MR 1738059 (2001c:20042)
- 5.
- J. Conway, R. T. Curtis, S. Norton, R. Parker, and R. Wilson, Atlas of finite groups, Clarendon Press, Oxford, 1985. MR 827219 (88g:20025)
- 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 (43:2113)
- 7.
- R. H. Dye, On the involution classes of the linear groups
 ,  ,  ,  over fields of characteristic two, Math. Proc. Cambridge Phil. Soc. 72 (1972), 1-6. MR 0294519 (45:3589)
- 8.
- R. H. Dye, On the conjugacy classes of involutions of the unitary groups
 ,  ,  ,  over perfect fields of characteristic  , J. Algebra 24 (1973), 453-459. MR 0308287 (46:7401)
- 9.
- K. Harada and M. Miyamoto, On the order of a group of even order, To appear in J. Algebra.
- 10.
- B. Huppert and N. Blackburn, Finite Groups III, Springer-Verlag, Berlin, 1982. MR 662826 (84i:20001b)
- 11.
- N. Iiyori and H. Yamaki, Prime graph components of the simple groups of Lie type over the fields of even characteristic, J. Algebra 155 (1993), 335-343. Corrigenda 181 (1996) 659. MR 1212233 (94e:05268)
- 12.
- A. S. Kondrat'ev, Prime graph components of finite simple groups, Math. USSR Sbornik 67 (1990), 235-247. MR 1015040 (90h:20018)
- 13.
- M. Suzuki, Group theory II, Springer-Verlag, Berlin, 1986. MR 815926 (87e:20001)
- 14.
- J. S. Williams, Prime graph components of finite groups, J. Algebra 69 (1981), 487-513. MR 617092 (82j:20054)
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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:
http://dx.doi.org/10.1090/S0002-9939-07-09118-6
PII:
S 0002-9939(07)09118-6
Keywords:
Finite simple groups,
centralizers of involutions
Received by editor(s):
August 15, 2006
Posted:
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.
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