Abstract
We study the relations between two groups related to cluster automorphism groups which are defined by Assem, Schiffler and Shamchenko. We establish the relationships among (strict) direct cluster automorphism groups and those groups consisting of periodicities of labeled seeds and exchange matrices, respectively, in the language of short exact sequences. As an application, we characterize automorphism-finite cluster algebras in the cases of bipartite seeds or finite mutation type. Finally, we study the relation between the group [Formula: see text] for a cluster algebra [Formula: see text] and the group [Formula: see text] for a mutation group [Formula: see text] and a labeled mutation class [Formula: see text], and we give a negative answer via counter-examples to King and Pressland's problem.
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