will be called an idempotent on G. The support of a complex function ƒ on G is the set of all # £ G at which ƒ(x) 5^0. The support group of ƒ is the smallest subgroup of G which contains the support of ƒ. By methods involving Fourier transforms and the Pontryagin duality theory, the idempotents on abelian groups are completely known [2, p. 199]. (For nondiscrete locally compact abelian groups, the classification of the idempotent measures was completed by P. J. Cohen [l].) Let us draw attention to the following facts, of which (A) and (D) are probably the most striking: (A) If ƒ is an idempotent on an abelian group G, then the support group of f is finite. (B) Idempotents on abelian groups are self-adjoint (i.e., fix") is the complex conjugate of ƒ(#)). (C) On a finite abelian group there are only finitely many idempotents (namely 2 if the group has n elements). On a countable abelian group there are at most countably many idempotents. (D) If ƒ is an idempotent on an abelian group and if ||/|| > 1, then | | / | |^>!\ /5 [3, p. 72]. (Note that there are no idempotents ƒ with jj/ll < 1 , except ƒ =0.) It is the purpose of the present note to show that each of the above statements becomes false if the word "abelian" is omitted. 1 Sponsored by the Mathematics Research Center, United States Army, Madison, Wisconsin under Contract No. DA-11-022-ORD-2059, and by NSF grant GP-249.