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arXiv · 2609.06755

Subgroups of free groups with the same commutator subgroup

Artur Tursunbaev
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Abstract

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.

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Artur Tursunbaev. 2026-09-06. Subgroups of free groups with the same commutator subgroup. https://arxiv.org/abs/2609.06755

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