On a decomposition of the Poisson distribution on groups

Let a locally compact abelian group $X=R^n\times G$, where $G$ contains a compact open subgroup $K$, $F$ is a finite measure on $X$ and
$$e(F)=\operatorname{exp}\{-F(X)\}\sum_{k=0}^\infty F^{\ast k}/k!$$
is a generalized Poisson distribution.
Theorem 1. {\it If $F(X)<1/2\ln 2$ and the measures $F^{\ast m}$ and $F^{\ast k}$ are mutually singular for any different integers $m$ and $k$ then $e(F)$ has no indecomposable divisors.}
Theorem 2. An absolutely continuous measure $F$ on $X$ such that $e(F)$ has no indecomposable divisors exists if and only if one of the following conditions is satisfied:
($\alpha$) $n=0$ and factor-group $G/K$ contains an element of infinite order,
($\beta$) $n>0$.