On Radon transforms on finite groups

TL;DR

This paper investigates whether functions on finite groups can be uniquely determined by sums over cyclic subgroup cosets, analyzing the Radon transform's injectivity using representation theory and providing conditions for injectivity and noninjectivity.

Contribution

It introduces a discrete inverse problem analogous to geodesic integrals on manifolds and characterizes the Radon transform's injectivity on finite groups, especially identifying Frobenius complements.

Findings

Radon transform is non-injective on Frobenius complements.

Provides conditions for injectivity and noninjectivity.

Complete characterization for abelian groups.

Abstract

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.