A reduction theorem for the Character Triple 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 extensively investigated by Dade in the 1990s. Our result shows that this role is played by the Character Triple Conjecture, recently introduced by Späth, which we present here in a general form free of all previously imposed restrictions. In particular, our result further highlights the Character Triple Conjecture as a statement of independent interest, rather than merely a condition that needs to be verified for quasi‐simple groups in order to obtain Dade's Conjecture. To conclude, we also show that the Character Triple Conjecture holds for every finite group with abelian Sylow 2‐subgroup with respect to any prime number.