A reduction theorem for the Character Triple Conjecture

TL;DR

This paper proves that the Character Triple Conjecture for all finite groups can be established if it holds for all quasi-simple groups, simplifying the approach to Dade's conjecture and advancing the understanding of group representation theory.

Contribution

The paper demonstrates a reduction theorem linking the Character Triple Conjecture's validity for all finite groups to its validity for quasi-simple groups, providing a self-reducing framework.

Findings

Character Triple Conjecture holds for all finite groups if true for quasi-simple groups.

Introduces a general, unrestricted form of the Character Triple Conjecture.

Answers a long-standing question about the self-reducing nature of Dade's conjecture.

Abstract

In this paper, we show that the Character Triple Conjecture holds for all finite groups once assumed for all quasi-simple groups. This answers the question on the existence of a self-reducing form of Dade's conjecture, a problem that was long investigated by Dade in the 1990s. Our result shows that this role is played by the Character Triple Conjecture, recently introduced by Sp\"ath, that we present here in a general form free of all previously imposed restrictions.