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In this article, we study the following Klein-Gordon-Maxwell system involving critical exponent (KGM){-△u+V(x)u-(2w+Φ)Φu=λ|u|^q-2u+|u|^4u,x∈R^3,-△Φ+Φu^2=-wu^2,x∈R^3,where λ and ω are two positive constants. We found the existence of positive ground state