  A notion of admissible probability measures $\mu$ on a locally compact
Abelian group (LCA-group) $G$ with connected dual group $\hat G=\R^d\times
\T^n$ is defined. To such a measure $\mu$, a closed semigroup
$\Lambda(\mu)\subseteq (0,\infty)$ can be associated, such that, for $t\in
\Lambda(\mu)$, the Fourier transform to the power $t$, $(\hat \mu)^t$, is a
characteristic function. We prove that the existence of roots for non
admissible probability measures underlies some restrictions, which do not hold
in the admissible case. As we show for the example $\Z_2$, in the case of
LCA-groups with non connected dual group, there is no canonical definition of
the set $\Lambda(\mu)$.
