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In this paper, we study relations between automorphism groups of cubic fourfolds and Kuznetsov components. Firstly, we characterize automorphism groups of cubic fourfolds as subgroups of autoequivalence groups of Kuznetsov components using Bridgeland stability conditions. Secondly, we compare automorphism groups of cubic fourfolds with automorphism groups of their associated K3 surfaces. Thirdly, we note that the existence of a non-trivial symplectic automorphism on a cubic fourfold is related to the existence of associated K3 surfaces.

Automorphism groups of cubic fourfolds and K3 categories