Abstract: Furstenberg's famous proof of Szemeredi's theorem leads to a natural question about the convergence and limit of some multiple ergodic averages. In the case of Z-actions these averages were studied by Host-Kra and Ziegler. They show that the limiting behavior of such multiple ergodic average is determined on a certain factor that can be given the structure of an inverse limit of nilsystems (i.e. rotations on a nilmanifold). This structure result can be generalized to Z^d actions (where the average is taken over a Folner sequence), but the non-finitely generated case is still open. The only progress prior to our work is due to Bergelson Tao and Ziegler, who studied actions of the infinite direct sum of Z/pZ. In our work we generalize this further to the case where the sum is taken over different primes (the most interesting case is when the multiset of primes is unbounded). We will explain how this case is significantly different from the work of Bergelson Tao and Ziegler by describing a new phenomenon that only happens in these settings. Moreover, we will discuss a generalized version of nilsystems that plays a role in our work and some corollaries. If time allows we will also discuss the group actions of other abelian groups