Generalized poisson distribution on groups

The article deals with the arithmetic of distributions on groups. Let $X$ be a locally compact separable abelian metric group, $e(F)=e^{-F(X)}(\varepsilon_0+F+\dfrac{F^{*2^-}}{z!}+\dots)$ the generalized Poisson distribution associated with a completely finite measure $F$, $I_0$ the class of distributions without indecomposable or idempotent divisors. Some conditions under which the generalized Poisson distributions belong and do not belong to the class $I_0$ are derived. Some other topics related to the class $I_0$ are considered.