A bound for $|G:\mathbf O_p(G)|_p$ in terms of the largest irreducible character degree of a finite $p$-solvable group $G$
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- by Diane Benjamin PDF
- Proc. Amer. Math. Soc. 127 (1999), 371-376 Request permission
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 $|P: {\mathbf {O}}_{p}(G)|\le p^{2\alpha }$.References
- I. Martin Isaacs, Character theory of finite groups, Pure and Applied Mathematics, No. 69, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. MR 0460423
- I. Martin Isaacs, Algebra, Brooks/Cole Publishing Co., Pacific Grove, CA, 1994. A graduate course. MR 1276273
- D. S. Passman, Groups with normal solvable Hall $p^{\prime }$-subgroups, Trans. Amer. Math. Soc. 123 (1966), 99–111. MR 195947, DOI 10.1090/S0002-9947-1966-0195947-2
Additional Information
- Diane Benjamin
- Affiliation: Department of Mathematics, University of Wisconsin – Platteville, Platteville, Wisconsin, 53818
- Email: benjamin@uwplatt.edu
- Received by editor(s): May 31, 1997
- Communicated by: Ronald M. Solomon
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 127 (1999), 371-376
- MSC (1991): Primary 20C15
- DOI: https://doi.org/10.1090/S0002-9939-99-04746-2
- MathSciNet review: 1485458