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 Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(e) ISSN 0002-9939(p)

     

Notes on the largest irreducible character degree of a finite group

Author(s): Qian Guohua
Journal: Proc. Amer. Math. Soc. 132 (2004), 1899-1903.
MSC (2000): Primary 20C15
Posted: January 23, 2004
MathSciNet review: 2053959
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Abstract: Let $G$ be a finite group and $b(G)$ the largest irreducible character degree of $G$. In this note, we show the following results: if $b(G)<p^2$, then $\vert G:O_p(G)\vert _p\leq p$ ; if $b(G)<p^m$ and, in addition, $G$ is $p$-solvable with abelian Sylow $p$-subgroup, then $ \vert G:O_p(G)\vert _p< p^m$.

References:

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D. Benjamin, A bound for $\vert G:O_p(G)\vert$ in terms of the largest irreducible character degree of a finite $p$-solvable group $G$, Proc. Amer. Math. Soc. 127:2 (1999), 371-376. MR 99c:20005

2.
J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, Atlas of finite groups, Oxford Univ. Press (Clarendon), Oxford and New York, 1985. MR 88g:20025

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D. Gorenstein, Finite simple groups: An introduction to their classification, Plenum Press, New York, 1982. MR 84j:20002

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I. M. Isaacs, Character theory of finite groups, Academic Press, New York, 1976. MR 57:417

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W. Willems, Block of defect zero in finite groups of Lie type, J. Algebra 113 (1988), 511-522. MR 89c:20025

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Additional Information:

Qian Guohua
Affiliation: Department of Mathematics, Changshu College, Changshu, Jiansu, 215500, People's Republic of China
Email: ghqian2000@yahoo.com.cn

DOI: 10.1090/S0002-9939-04-07316-2
PII: S 0002-9939(04)07316-2
Keywords: Finite group, character degree
Received by editor(s): January 28, 2003
Received by editor(s) in revised form: April 3, 2003
Posted: January 23, 2004
Additional Notes: Project supported by the National Nature Science Foundation of China and the Nature Science Foundation of Jiangsu Provincial Education Department (03KJB11002).
Communicated by: Stephen D. Smith
Copyright of article: Copyright 2004, American Mathematical Society




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