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countable free group

Source: miklos schweitzer 2005 q4

August 12, 2021
abstract algebrafree group

Problem Statement

Let F be a countable free group and let F=H1>H2>H3>⋯F = H_1> H_2> H_3> \cdots be a descending chain of finite index subgroups of group F. Suppose that ∩Hi\cap H_i does not contain any nontrivial normal subgroups of F. Prove that there exist gi∈Fg_i\in F for which the conjugated subgroups HigiH_i^{g_i} also form a chain, and ∩Higi={1}\cap H_i^{g_i}=\{1\}.
Nielsen-Schreier Theorem might be useful.
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