Subgroups of free groups with the same commutator subgroup

RESUMO

We prove that if H and K are subgroups of a free group F of arbitrary rank and $H'=K'\ne 1$, then $H=K$. As a consequence, if R is a noncyclic subgroup of a free group F and $R'$ is normal in F, then R is normal in F. This gives a positive answer to a problem of Shpilrain, originally posed as Problem 11.124 in the Kourovka Notebook and later appearing as Problem F15 in Kapovich-Myasnikov-Shpilrain.