How close can we come to a parity function when there isn't one?

Consider a group G such that there is no homomorphism f : G ! f� 1g. In that case, how close can we come to such a homomorphism? We show that if f has zero expectation, then the probability that f(xy) = f(x)f(y), where x;yare chosen uniformly and independently from G, is at most 1=2(1 + 1= p d), where d is the dimension of G’s smallest nontrivial irreducible representation. For the alternating group An, for instance, d = n − 1. On the other hand, An contains a subgroup isomorphic to Sn 2, whose parity function we can extend to obtain an f for which this probability is 1=2(1 + 1= n � ). Thus the extent to which f can be “more homomorphic” than a random function from An to f� 1g lies between O(n 1/2 ) and (n 2 ).