Simple groups of order
Author:
Leo J. Alex
Journal:
Trans. Amer. Math. Soc. 173 (1972), 389399
MSC:
Primary 20D05
MathSciNet review:
0318291
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Abstract: Let denote the projective special linear group of degree over , the field with elements. The following theorem is proved. Theorem. Let be a simple group of order an odd prime. If the index of a Sylow subgroup of in its normalizer is two, then is isomorphic to one of the groups, , and .
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 [1]
 Leo J. Alex, On simple groups of order , J. Algebra (to appear).
 [2]
 , Some exponential Diophantine equations which arise in finite groups (to appear).
 [3]
 H. L. Blichfeldt, Finite collineation groups, Univ. of Chicage Press, Chicago, Ill., 1917.
 [4]
 Richard Brauer, On simple groups of order , Bull. Amer. Math. Soc. 74 (1968), 900903. MR 38 #4552. MR 0236255 (38:4552)
 [5]
 , On groups whose order contains a prime number to the first power. I, II, Amer. J. Math. 64 (1942), 401440. MR 4, 1; 2. MR 0006537 (4:1e)
 [6]
 , Über endliche lineare Gruppen von Primzahlgrad, Math. Ann. 169 (1967), 7396. MR 34 #5913. MR 0206088 (34:5913)
 [7]
 Richard Brauer and K. A. Fowler, On groups of even order, Ann. of Math. (2) 62 (1955), 565583. MR 17, 580. MR 0074414 (17:580e)
 [8]
 Richard Brauer and M. Suzuki, On finite groups of even order whose Sylow subgroup is a quaternion group, Proc. Nat. Acad. Sci. U.S.A. 45 (1959), 17571759. MR 22 #731. MR 0109846 (22:731)
 [9]
 Richard Brauer and H. F. Tuan, On simple groups of finite order, Bull. Amer. Math. Soc. 51 (1945), 756766. MR 7, 371. MR 0015102 (7:371d)
 [10]
 Walter Feit, The current situation in the theory of finite simple groups, Mimeographed notes, Yale University, New Haven, Conn., 1970. MR 0427449 (55:481)
 [11]
 , On finite linear groups, J. Algebra 5 (1967), 378400. MR 34 #7632. MR 0207818 (34:7632)
 [12]
 Daniel Gorenstein, Finite groups in which Sylow subgroups are abelian and centralizers of involutions are solvable, Canad. J. Math. 17 (1965), 860906. MR 32 #7635. MR 0190221 (32:7635)
 [13]
 Daniel Gorenstein and J. H. Walter, The characterization of finite groups with dihedral Sylow subgroups. I, II, III, J. Algebra 2 (1965), 85151; 218270; 354393. MR 31 #1297a, b; MR 32 #7634.
 [14]
 D. H. Lehmer, On a problem of Störmer, Illinois J. Math. 8 (1964), 5779. MR 28 #2072. MR 0158849 (28:2072)
 [15]
 David Mutchler, On simple groups of order (unpublished).
 [16]
 Michio Suzuki, Finite groups in which the centralizer of any element of order 2 is closed, Ann. of Math. (2) 82 (1965), 191212. MR 32 #1250. MR 0183773 (32:1250)
 [17]
 David B. Wales, Finite linear groups of degree seven. II, Pacific J. Math. 34 (1970), 207235. MR 42 #1918. MR 0267016 (42:1918)
 [18]
 , Simple groups of order , J. Algebra 16 (1970), 575596. MR 42 #3169. MR 0268270 (42:3169)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197203182911
PII:
S 00029947(1972)03182911
Keywords:
Finite simple group classification,
class algebra coefficient,
characters of finite groups,
principal block
Article copyright:
© Copyright 1972
American Mathematical Society
