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Let $\mathcal{P}$ be a countable multiset of primes and let$G=\bigoplus_{p\in P}\mathbb{Z}/p\mathbb{Z}$. We study the universalcharacteristic factors associated with the Gowers-Host-Kra seminorms for thegroup $G$. We show that the universal characteristic factor of order $<k+1$ isa factor of an inverse limit of finite dimensional $k$-step nilpotenthomogeneous spaces. The latter is a counterpart of a $k$-step nilsystem wherethe homogeneous group is not necessarily a Lie group. This result provides acounterpart of the structure theorem of Host-Kra and Ziegler concerning$\mathbb{Z}$-actions and generalizes the results of Bergelson Tao and Zieglerconcerning $\mathbb{F}_p^\omega$-actions. This result is the first instance ofa structure theorem for the universal characteristic factors associated with anon-finitely generated group of unbounded torsion. As an application we derivean alternative proof for the $L^2$-convergence of multiple ergodic averagesassociated with $k$-term arithmetic progressions in $G$ and derive a formulafor the limit in the special case where the underlying space is a nilpotenthomogeneous system.

Host-Kra factors for $\bigoplus_{p\in P}\mathbb{Z}/p\mathbb{Z}$ actions and finite dimensional nilpotent systems