The Alperin-McKay Conjecture for metacyclic, minimal non-abelian defect groups

TL;DR

This paper proves the Alperin-McKay Conjecture for specific metacyclic, minimal non-abelian defect groups in finite groups, including cases for p=3 and p=5, without relying on the classification of finite simple groups.

Contribution

It establishes the conjecture for a new class of defect groups, expanding the understanding of block theory in finite groups.

Findings

Proves Alperin-McKay Conjecture for metacyclic, minimal non-abelian defect groups.

Verifies Alperin's Weight Conjecture for p=3 in these cases.

Extends results to certain non-abelian defect groups at p=5.

Abstract

We prove the Alperin-McKay Conjecture for all $p$-blocks of finite groups with metacyclic, minimal non-abelian defect groups. These are precisely the metacyclic groups whose derived subgroup have order $p$. In the special case $p=3$, we also verify Alperin's Weight Conjecture for these defect groups. Moreover, in case $p=5$ we do the same for the non-abelian defect groups $C_{25}\rtimes C_{5^n}$. The proofs do not rely on the classification of the finite simple groups.