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Bachet elliptic curves are the curves y^2=x^3+a^3 and in this work the groupstructure E(F_{p}) of these curves over finite fields F_{p} is considered. Itis shown that there are two possible structures E(F_{p}){\cong}C_{p+1} orE(F_{p}){\cong}C_{n}{\times}C_{nm}, for m,n{\in}{\mathbb{N}}, according top{\equiv}5 (mod6) and p{\equiv}1 (mod6), respectively. A result of Washingtonis restated in a more specific way saying that ifE(F_{p}){\cong}Z_{n}{\times}Z_{n}, then p{\equiv}7 (mod12) and p=n^2{\mp}n+1.

The Group Structure of Bachet Elliptic Curves over Finite Fields F_{p}