Some remarks on the Pompeiu problem for groups

A Borel set E in a topological group G is said to be a P-set for the space of integrable functions on G if the zero function is the only integrable function whose integral over all left and right translates of E by elements of G is zero. For a "sufficiently nice" group G and a Borel set E of finite Haar measure a certain condition on the Fourier transform of a function related to E is shown to be a sufficient condition for E to be a P-set. This condition is then applied to several classes of groups including certain compact groups, certain semisimple Lie groups, the Heisenberg groups and the Euclidean motion group of the plane