Galois's lost proof on $PSL(2,p)$

Abstract

In his famous ``testamentary letter'' Galois formulated several results without proof. Among them he claimed --- here we express it in modern terms --- that the projective special linear group $PSL(2,p)$ has a subgroup of index $p$ only for $p\le 11$. Nowadays this result is seen as a corollary of the complete enumeration of all subgroups of $PSL(2,p)$ obtained by Gierster in 1881. In this paper we attempt to give a proof that uses only methods available to Galois.