  A result of the author shows that the behavior of Gowers norms on bounded
exponent abelian groups is connected to finite nilspaces. Motivated by this, we
investigate the structure of finite nilspaces. As an application we prove
inverse theorems for the Gowers norms on bounded exponent abelian groups. It
says roughly speaking that if a function on A has non negligible U(k+1)-norm
then it correlates with a phase polynomial of degree k when lifted to some
abelian group extension of A. This result is closely related to a conjecture by
Tao and Ziegler. In prticular we obtain a new proof for the Tao-Ziegler inverse
theorem.
