Finiteness of subgroups of a fixed finite index
= Finiteness of subgroups of a fixed finite index
A <finitely generated group> has only finitely many <subgroups> of any prescribed <index of a subgroup> $n$. Every such subgroup is a point stabilizer for a transitive action on $n$ points, while a homomorphism from a finitely generated group to the finite <symmetric group> $S_n$ has only finitely many possible images of the generators.