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A projective non-singular plane algebraic curve of degree d<=4 is calledmaximally symmetric if it attains the maximum order of the automorphism groupsfor complex non-singular plane algebraic curves of degree d. For d<=7, all suchcurves are known. Up to projectivities, they are the Fermat curve for d=5,7,see \cite{kmp1,kmp2}, the Klein quartic for d=4, see \cite{har}, and the Wimansextic for d=6, see \cite{doi}. In this paper we work on projective planecurves defined over an algebraically closed field of characteristic zero, andwe extend this result to every d>=8 showing that the Fermat curve is the uniquemaximally symmetric non-singular curve of degree d with d>=8, up toprojectivity. For d=11,13,17,19, this characterization of the Fermat curve hasalready been obtained, see \cite{kmp2}.

The Fermat curve x^n+y^n+z^n: the most symmetric non-singular algebraic plane curve