  We study a method of generating Bell inequalities by using group actions of
single-generator abelian groups. Two parties, Alice and Bob, each make one of M
possible measurements on a system, with each measurement having K possible
outcomes. The probabilities for the outcomes of these measurements are P(a_j =
k, b_{j'}=k'), where j,j' are in the set {1,2,... M} and k,k' are in the set
{0,1,... K-1}. The sums of some subsets of these probabilities have upper
bounds when the probabilities result from a local, realistic theory that can be
violated if the probabilities come from quantum mechanics. In our case the
subsets of probabilities are generated by a group action, in particular, a
representation of a single-generator group acting on product states in a
tensor-product Hilbert space. We show how this works for several cases,
including M=2, K=3, and general M, K=2. We also discuss the resulting
inequalities in terms of nonlocal games.
