# Abelian subgroups, nilpotent subgroups, and the largest character degree of a finite group

## TL;DR

This paper establishes bounds relating abelian and nilpotent subgroups of finite groups to the largest character degree, providing new insights into the structure of finite groups.

## Contribution

It proves that the size of certain subgroup quotients is bounded by the largest character degree, extending known results to abelian and nilpotent subgroups.

## Key findings

- Bound on |H O_{π}(G)/ O_{π}(G)| by the largest character degree
- Similar bounds established for nilpotent subgroups
- Results deepen understanding of subgroup structure in finite groups

## Abstract

Let $H$ be an abelian subgroup of a finite group $G$ and $\pi$ the set of prime divisors of $|H|$. We prove that $|H O_{\pi}(G)/ O_{\pi}(G)|$ is bounded above by the largest character degree of $G$. A similar result is obtained when $H$ is nilpotent.