NOTES ON AUTOMORPHISMS OF SURFACES OF GENERAL TYPE WITH AND

Abstract

Let$S$be a smooth minimal complex surface of general type with$p_{g}=0$and$K^{2}=7$. We prove that any involution on$S$is in the center of the automorphism group of$S$. As an application, we show that the automorphism group of an Inoue surface with$K^{2}=7$is isomorphic to$\mathbb{Z}_{2}^{2}$or$\mathbb{Z}_{2}\times \mathbb{Z}_{4}$. We construct a$2$-dimensional family of Inoue surfaces with automorphism groups isomorphic to$\mathbb{Z}_{2}\times \mathbb{Z}_{4}$.