Bounding an index by the largest character degree of a solvable group

Mark L. Lewis

Bounding an index by the largest character degree of a solvable group

Group Theory 2011-11-16 v2

Authors: Mark L. Lewis

Abstract

In this paper, we show that if $p$ is a prime and $G$ is a $p$ -solvable group, then $| G:O_p (G) |_p \le (b(G)^p/p)^{1/(p-1)}$ where $b(G)$ is the largest character degree of $G$ . If $p$ is an odd prime that is not a Mersenne prime or if the nilpotence class of a Sylow $p$ -subgroup of $G$ is at most $p$ , then $| G:O_p (G) |_p \le b(G)$ .

Keywords

solvable group character degree p-solvable group Sylow p-subgroup nilpotence class Mersenne prime

Cite

@article{arxiv.1009.5434,
  title  = {Bounding an index by the largest character degree of a solvable group},
  author = {Mark L. Lewis},
  journal= {arXiv preprint arXiv:1009.5434},
  year   = {2011}
}

Comments

Updated the introduction based on referee's report

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