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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A bound for $|G:{\mathbf{O}}_{p}(G)|_{p}$ in terms
of the largest irreducible character degree
of a finite $p$-solvable group $G$


Author: Diane Benjamin
Journal: Proc. Amer. Math. Soc. 127 (1999), 371-376
MSC (1991): Primary 20C15
MathSciNet review: 1485458
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Abstract: Let $b(G)$ denote the largest irreducible character degree of a finite group $G$, and let $p$ be a prime. Two results are obtained. First, we show that, if $G$ is a $p$-solvable group and if $b(G) < p^{2}$, then $p^{2} {\not \big \vert }\,|\,G:{\mathbf{O}}_{p}(G)|$. Next, we restrict attention to solvable groups and show that, if $b(G) \le p^{\alpha }$ and if $P$ is a Sylow $p$-subgroup of $G$, then $|P: {\mathbf{O}}_{p}(G)|\le p^{2\alpha }$.


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Additional Information

Diane Benjamin
Affiliation: Department of Mathematics, University of Wisconsin – Platteville, Platteville, Wisconsin, 53818
Email: benjamin@uwplatt.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-99-04746-2
PII: S 0002-9939(99)04746-2
Received by editor(s): May 31, 1997
Communicated by: Ronald M. Solomon
Article copyright: © Copyright 1999 American Mathematical Society