  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.
