ABSTRACT: Høyer has given a generalisation of the Deutsch-Jozsa algorithm which uses the Fourier transform on a group G which is (in general) non-Abelian. His algorithm distinguishes between functions which are either perfectly balanced (m-to-one) or constant, with certainty, and using a single quantum query. Here, we show that this algorithm (which we call the Deutsch-Jozsa-Høyer algorithm) can in fact deal with a broader range of promises, which we define in terms of the irreducible representations of G.