On Translations of Sets in Topological Groups

Various authors {[5], [6], [7]} have obtained results on transformations of sets in the n-dimensional Euclidean space. In this paper, we prove some of these results in a topological group. In proving one of the results {Theorem 5.2}, we require a theorem on density of sets in a topological group which we prove with the help of a Vitali type theorem as proved in [1] for invariant measure. The theorem on density of sets {Theorem 5.1} has some interest in itself. The notion of density of sets has been extended from classical one [4] to a metric space [2] and to a measure space [8] and Romanovski space [10] with the help of a Vitali type theorem in a measure space [9].