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Let $\Gamma$ be a countable abelian group, let $k\geq 1$, and let$\mathrm{X}=(X,\mathcal{X},\mu,T)$ be an ergodic $\Gamma$-system of order $k$in the sense of Host--Kra--Ziegler. The $\Gamma$-system $\mathrm{X}$ is said tobe totally disconnected if all its structure groups are totally disconnected.We show that any totally disconnected $\Gamma$-system of order $k$ is ageneralized factor of a $\mathbb{Z}^\omega$-system with the structure of a Weylsystem. As a consequence of this structure theorem, we show that totallydisconnected $\Gamma$-systems of order $k$ are represented by translations ondouble cosets of nilpotent Polish groups. By a correspondence principle of twoof us, we can use this representation to establish a (weak) inverse theorem forthe $U^k$ Gowers uniformity norms on finite abelian groups of bounded torsion.

The structure of totally disconnected Host--Kra--Ziegler factors, and  the inverse theorem for the $U^k$ Gowers uniformity norms on finite abelian  groups of bounded torsion