On decomposition of a convolution of two Poisson distributions on locally compact Abelian groups

Let $X$ be a locally compact Abelian separable group, $\displaystyle\pi=\operatorname{exp}\{-F(x)\}\sum_{n=0}^{\infty}F^{\ast n/n!}$ be the Poisson distribution (P. d.) on $X$ generated by the positive measure $F$ concentrated in the point $x\in X$. It is shown in the paper that if the elements $x_1$ and $x_2$ generating P. d.'s $\pi_1$ and $\pi_2$ have infinite order, then every divisor of the convolution $\mu=\pi_1\ast\pi_2$ is a shift of the convolution of two P. d.'s.