On a covering group theorem and its applications

Let $p\colon X\to G$ be an n-fold covering of a compact group $G$ by a connected topological space $X$ Then there exists a group structure in $X$ turning $p$ into a homomorphism between compact groups. As an application, we describe all $n$-fold coverings of a compact connected abelian group. Also, a criterion of triviality for $n$-fold coverings in terms of the dual group and the one-dimensional Čech cohomology group is obtained.