There are two basic theorems on arithmetic properties of probability measures on Euclidean space: the Levy decomposition of infinitely divisible probability measures as convolutions of Poisson and Gaussian measures, and the Khintchine factorization of arbitrary probability measures in terms of indecomposable measures and measures without indecomposable factors. Both theorems have been generalized by K. R. Parthasarathy to measures on an Abelian locally compact group. Within this framework the role of Gaussian factors will be discussed. Moreover, characterizations of Gaussian measures (in the sense of Cramer and Bernstein) will be presented whose validity depends on the structure of the underlying group.