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On The Largest Character Degree And Solvable Subgroups Of Finite Groups

Zongshu Wu, Yong Yang

Abstract

Let $G$ be a finite group, and $π$ be a set of primes. The $π$-core $\mathbf{O}_π(G)$ is the unique maximal normal $π$-subgroup of $G$, and $b(G)$ is the largest irreducible character degree of $G$. In 2017, Qian and Yang proved that if $H$ is a solvable $π$-subgroup of $G$, then $|H\mathbf{O}_π(G)/\mathbf{O}_π(G)|\le b(G)^3$. In this paper, we improve the exponent of $3$ to $3\log_{504}(168)<2.471$.

On The Largest Character Degree And Solvable Subgroups Of Finite Groups

Abstract

Let be a finite group, and be a set of primes. The -core is the unique maximal normal -subgroup of , and is the largest irreducible character degree of . In 2017, Qian and Yang proved that if is a solvable -subgroup of , then . In this paper, we improve the exponent of to .

Paper Structure

This paper contains 6 sections, 7 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Bounds on Finite Simple Groups
  3. Permutation Groups and Colorings
  4. Proof of the Main Result
  5. Further Discussions
  6. Acknowledgements

Key Result

Theorem 1.1

Let $H$ be a solvable $\pi$-subgroup of $G$. Then $|H\mathbf O_\pi(G)/\mathbf O_\pi(G)|<b(G)^3$.

Theorems & Definitions (7)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Theorem 3.1
  • Corollary 3.2
  • Lemma 4.1
  • Lemma 4.2