In this paper we investigate when the generic member of a family of complex K3 surfaces admitting a non-symplectic automorphism of finite order admits also a symplectic automorphism of the same order. We give a complete answer to this question if the order of the automorphism is a prime number and we provide several examples and partial results otherwise. Moreover we prove that, under certain conditions, a K3 surface admitting a non-symplectic automorphism of prime odd order, , also admits a non-symplectic automorphism of order . This generalizes a previous result by J. Dillies for .
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Alice Garbagnati, Alessandra Sarti, On symplectic and non-symplectic automorphisms of K3 surfaces. Rev. Mat. Iberoam. 29 (2013), no. 1, pp. 135–162DOI 10.4171/RMI/716