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 $P({a}_{j}=k,{b}_{{j}^{\ensuremath{'}}}={k}^{\ensuremath{'}})$, where $j,{j}^{\ensuremath{'}}\ensuremath{\in}{1,2,...,M}$ and $k,{k}^{\ensuremath{'}}\ensuremath{\in}{0,1,...,K\ensuremath{-}1}$. 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.