The group theoretic interpretation of harmonic analysis was born out of the observation that the discrete Fourier transform on a signal of length $n$ was precisely the Fourier transform of the finite group $\mathbb{Z}/n\mathbb{Z}.$ I was thinking about the extent to which this analogy holds, but I was having trouble with the Laplacian, which, of course, occupies a central role in harmonic analysis. In particular, there doesn't seem to be a generalization of the operator to finite groups (though certainly there is for compact connected Lie groups).

Does anyone know of such an object?