  Finite-sheeted covering mappings onto compact connected groups are studied.
It is shown that a finite-sheeted covering mapping from a connected Hausdorff
topological space onto a compact connected abelian group G must be a
homeomorphism provided that the character group of G admits division by the
degree of given covering mapping. Using this result, we obtain criteria of
triviality for finite coverings of G in terms of its character group and means
on G. In order to establish these facts, for a k-sheeted covering mapping from
a compact topological space onto a compact connected group, we construct an
inverse system of k-sheeted coverings onto Lie groups which approximates this
covering mapping and prove a covering group theorem.
