  In this work we are going to study the dynamics of the linear automorphisms
of a measure convolution algebra over a finite group, $T(\mu)=\nu * \mu$. In
order to understand an classify the asymptotic behavior of this dynamical
system we provide an alternative to classical results, a very direct way to
understand convergence of the sequence $\{\nu^{n}\}_{n\in\mathbb{N}}$, where
$G$ is a finite group, $\nu\in\mathcal{P}(G)$ and
$\nu^n=\underbrace{\nu\ast...\ast\nu}_n$, trough the subgroup generated by his
support.
