According to M. Meo, Cauchy's proof of Cauchy's theorem (existence of elements of order a given prime p in every finite group of order divisible by a p) is wrong.
Cauchy works with subgroups of $S_n$, and his proof depends on the construction of what we now call a Sylow subgroup of $S_n$. This subgroup is obtained as a semidirect product, which Cauchy seems to say is actually a direct product (which would be abelian). I am not completely sure whether Cauchy was really wrong, or he did know what was going on, and simply lacked the appropriate language. In any case, would be an example of Lack of foundations.
