Periodicities in cluster algebras and cluster automorphism groups

In this paper, we study the relations between groups related to cluster automorphism groups which are defined by Assem, Schiffler and Shamchenko in \cite{ASS}. We establish the relationship among (strict) direct cluster automorphism groups and those groups consisting of periodicities of respectively labeled seeds and exchange matrices in the language of short exact sequences. As an application, we characterize automorphism-finite cluster algebras in the cases with bipartite seeds or finite mutation type. Finally, we study the relation between the groups $\mathrm{Aut}\mathcal{A}$ and $\mathrm{Aut}_{M_n}S$ and give the negative answer via counter-examples to King and Pressland's a problem in \cite{KP}.