Let X be a locally compact Abelian separable metric group, \(e(F)=\exp\{- F(X)\}(E_ 0+F+((F^{*2}/2!)+...)\) be the generalized Poisson distribution associated with a finite measure F and \(I_ 0\) be a class of distributions without indecomposable or idempotent divisors. Theorem 1. If the measures \(F^{*n}\) and \(F^{*m}\) are mutually singular for any different integers n and m, then \(e(F)\in I_ 0.\) Theorem 2. If a group X is nondiscrete, then the class \(I_ 0\) is dense in the set of all infinitely divisible distributions. Theorem 3. For every infinitely divisible distribution of a group X to be representable either as finite or infinite convolution of the distributions of the class \(I_ 0\) it is necessary and sufficient that the group X is isomorphic to \(R^ n+D\), \(n\geq 0\), D is a discrete group which does not contain elements of a finite order \(p\neq 2\).