The character triple conjecture for height zero characters and the prime $2$

Abstract

We prove that Späth’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 (one implication of) Brauer’s height zero conjecture. We also obtain similar results for the block-free version of the character triple conjecture at any prime $p$.