A block theoretic proof of Thompson's $A\times B$-Lemma

TL;DR

This paper presents a new proof of Thompson's $A\times B$-Lemma using the Brauer pair version of Brauer's Third Main Theorem, linking two important results in group theory.

Contribution

It provides a block theoretic proof of Thompson's $A\times B$-Lemma, connecting it to Brauer's Third Main Theorem for the first time.

Findings

Thompson's $A\times B$-Lemma can be derived from Brauer's Third Main Theorem.

Establishes a new proof technique using block theory.

Bridges concepts between local group theory and block theory.

Abstract

We show that Thompson's $A\times B$-Lemma can be obtained as a consequence of the Brauer pair version of Brauer's Third Main Theorem.