The Character Triple Conjecture for maximal defect characters and the prime 2

TL;DR

This paper proves the Character Triple Conjecture for finite groups at prime 2 with maximal defect characters, leveraging recent advances in the inductive Alperin-McKay condition, and extends results to abelian defect groups and other primes.

Contribution

It establishes the conjecture for all 2-blocks with abelian defect groups by reducing to the inductive Alperin-McKay condition and applying Brauer's Height Zero Conjecture.

Findings

Proves the Character Triple Conjecture for all finite groups at prime 2 with maximal defect characters.

Reduces the conjecture to the inductive Alperin-McKay condition, verified by Ruhstorfer.

Extends results to 2-blocks with abelian defect groups and block-free versions at any prime p.

Abstract

We prove that Sp\"ath's Character Triple Conjecture holds for every finite group with respect to maximal defect characters at the prime 2. This is done by reducing the maximal defect case of the conjecture to the so-called inductive Alperin-McKay condition whose verification has recently been completed by Ruhstorfer for the prime 2. As a consequence we obtain the Character Triple Conjecture for all 2-blocks with abelian defect groups by applying Brauer's Height Zero Conjecture, a proof of which is now available. We also obtain similar results for the block-free version of the Character Triple Conjecture at any prime p.