Under the degree of some finite linear groups
Author:
Harvey I. Blau
Journal:
Trans. Amer. Math. Soc. 155 (1971), 95113
MSC:
Primary 20.80
Erratum:
Trans. Amer. Math. Soc. 162 (1971), 475.
MathSciNet review:
0274604
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Abstract: Let be a finite group with a cyclic Sylow subgroup for some prime . Assume that is not of type , and that has a faithful indecomposable modular representation of degree . This paper offers several improvements of the known bound . In particular, . Other bounds are given relative to the order of the center of and the index of the centralizer of in its normalizer.
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 1.
 H. Blau, On the degree of some finite linear groups, Ph.D. Thesis, Yale University, New Haven, Conn., 1969.
 [1]
 R. Brauer, Investigations on group characters, Ann. of Math. (2) 42 (1941), 936958. MR 3, 196. MR 0005731 (3:196b)
 [2]
 , On groups whose order contains a prime number to the first power. I, II, Amer. J. Math. 64 (1942), 421440. MR 4, 1; MR 4, 2. MR 0006538 (4:2a)
 [3]
 , Some results on finite groups whose order contains a prime to the first power, Nagoya Math. J. 27 (1966), 381399. MR 33 #7402. MR 0199253 (33:7402)
 [4]
 E. C. Dade, Blocks with cyclic defect groups, Ann. of Math. (2) 84 (1966), 2048. MR 34 #251. MR 0200355 (34:251)
 [5]
 W. Feit, Groups with a cyclic Sylow subgroup, Nagoya Math. J. 27 (1966), 571584. MR 33 #7404. MR 0199255 (33:7404)
 [6]
 , On finite linear groups, J. Algebra 5 (1967), 378400. MR 34 #7632. MR 0207818 (34:7632)
 [7]
 , Modular representations of finite groups, Lecture Notes, Yale University, New Haven, Conn., 1969.
 [8]
 J. A. Green, On the indecomposable representations of a finite group, Math. Z. 70 (1958/59), 430445. MR 24 #A1304. MR 0131454 (24:A1304)
 [9]
 , Blocks of modular representations, Math. Z. 79 (1962), 100115. MR 25 #5114. MR 0141717 (25:5114)
 [10]
 , The modular representation algebra of a finite group, Illinois J. Math. 6 (1962), 607619. MR 25 #5106. MR 0141709 (25:5106)
 [11]
 Z. Janko, A new finite simple group with abelian Sylow subgroups and its characterization, J. Algebra 3 (1966), 147186. MR 33 #1359. MR 0193138 (33:1359)
 [12]
 G. J. Janusz, Indecomposable modules for finite groups, Ann. of Math. (2) 89 (1969), 209241. MR 39 #5622. MR 0244307 (39:5622)
 [13]
 B. Rothschild, Degrees of irreducible modular characters of blocks with cyclic defect groups, Bull. Amer. Math. Soc. 73 (1967), 102104. MR 34 #4381. MR 0204542 (34:4381)
 [14]
 J. G. Thompson, Vertices and sources, J. Algebra 6 (1967), 16. MR 34 #7677. MR 0207863 (34:7677)
 [15]
 H. F. Tuan, On groups whose orders contain a prime number to the first power, Ann. of Math. (2) 45 (1944), 110140. MR 5, 143. MR 0009397 (5:143f)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002994719710274604X
PII:
S 00029947(1971)0274604X
Keywords:
Indecomposable modular representation,
small degree,
cyclic Sylow subgroup,
symmetric decomposition,
skew decomposition,
irreducible complex representation
Article copyright:
© Copyright 1971
American Mathematical Society
