Note on a theorem of Schreier

Abstract

Schreier [2] showed (using graph techniques) that if F is a free group and H is a normal subgroup which is finitely generated, then H must be of finite index in F. We give an algebraic proof of the following generalization: Theorem 1.Let F be a free group on the free generators {a,} and let H be a finitely generated subgroup containing a normal subgroup of F.