ar X iv : m at h / 02 08 02 6 v 1 [ m at h . G R ] 4 A ug 2 00 2 A new proof of a theorem of Ivanov and Schupp

The object of this note is to give a very short proof of the following theorem of Ivanov and Schupp. Let H be a finitely generated subgroup of a free group F and the index [F : H ] infinite. Then there exists a nontrivial normal subgroup N of F such that N ∩ H = {1}. As noted in the abstract above the object of this note is to give a new proof of the following theorem of S.V.Ivanov and P.E.Schupp [2] Theorem 0.1. Let H be a finitely generated subgroup of a free group F and the index [F : H ] infinite. Then there exists a nontrivial normal subgroup N of F such that N ∩ H = {1}. Our proof depends on two facts. The first is a theorem of M. Hall [1] which states that a finitely generated subgroup of a finitely generated free group is a free factor of a subgroup of finite index. The second key fact, is the following simple lemma. Lemma 0.2. Let I be a normal subgroup of finite index of a group F , and let L be a normal subgroup of I. Furthermore let b1, ..., bm be a complete set of representatives of the right cosets Ig of I in F . Then