Let F be a countable free group and let F=H1>H2>H3>⋯ be a descending chain of finite index subgroups of group F. Suppose that ∩Hi does not contain any nontrivial normal subgroups of F. Prove that there exist gi∈F for which the conjugated subgroups Higi also form a chain, and ∩Higi={1}.Nielsen-Schreier Theorem might be useful. abstract algebrafree group