Finite groups contain large centralizers

TL;DR

This paper proves that in any finite non-abelian group, there exists a non-central element with a centralizer larger than the cube root of the group's order, using the Feit-Thompson theorem.

Contribution

It establishes a universal lower bound on the size of centralizers of non-central elements in finite groups without relying on classification.

Findings

Existence of a non-central element with large centralizer in finite groups

Centralizer size exceeds the cube root of the group order for some element

Proof avoids classification of finite simple groups

Abstract

Every finite non-abelian group of order $n$ has a non-central element whose centralizer has order exceeding $n^{1/3}$. The proof does not rely on the classification of finite simple groups, yet it uses the Feit-Thompson theorem.