# Is there any way to generalize the Laplacian to finite groups?

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?

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Take the Laplacian of a Cayley graph for the group. –  Qiaochu Yuan May 31 '12 at 19:14
When you say compact groups you mean compact connected Lie groups, right? Finite groups are compact, after all. –  Qiaochu Yuan May 31 '12 at 19:14
@Qiaochu Indeed, thanks. –  Grant Rotskoff Jun 1 '12 at 2:12