On Radon transforms on finite groups

If $G$ is a finite group, is a function $f:G\to\mathbb C$ determined by its sums over all cosets of cyclic subgroups of $G$? In other words, is the Radon transform on $G$ injective? This inverse problem is a discrete analogue of asking whether a function on a compact Lie group is determined by its integrals over all geodesics. We discuss what makes this new discrete inverse problem analogous to well-studied inverse problems on manifolds and we also present some alternative definitions. We use representation theory to prove that the Radon transform fails to be injective precisely on Frobenius complements. We also give easy-to-check sufficient conditions for injectivity and noninjectivity for the Radon transform, including a complete answer for abelian groups and several examples for nonabelian ones.

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