Using a tensorial approach, we show how to construct a one-one correspondence between pattern probabilities and edge parameters for any group-based model. This is a generalisation of the "Hadamard conjugation" and is equivalent to standard results that use Fourier analysis. In our derivation we focus on the connections to group representation theory and emphasize that the inversion is possible because, under their usual definition, group-based models are defined for abelian groups only. We also argue that our approach is elementary in the sense that it can be understood as simple matrix multiplication where matrices are rectangular and indexed by ordered-partitions of varying sizes.