We extend an uncertainty principle due to Cowling and Price to Euclidean spaces, Heisenberg groups and the Euclidean motion group of the plane. This uncertainty principle is a generalisation of a classical result due to Hardy. We also show that on the real line this uncertainty principle is almost equivalent to Hardy's theorem.1991 Mathematics subject classification (Amer. Math. Soc): primary 22E25, 22E30.

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It is proved that if G is a compact connected semisimple Lie group and H a compact group of homeomorphisms of G containing all left and right translations of G, then there exists a positive integer k such that for any t s H, t* is, modulo a translation, an inner automorphism.