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Subgroups of free groups with the same derived subgroup
Abstract
We prove that if H and K are subgroups of a free group F of arbitrary rank and H'=K'≠ 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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BibTeX
@misc{tursunbaev2026subgroups,
title = {{Subgroups of free groups with the same derived subgroup}},
author = {Artur Tursunbaev},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06755},
eprint = {2609.06755},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06755},
}
RIS
TY - UNPB
TI - Subgroups of free groups with the same derived subgroup
AU - Artur Tursunbaev
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.06755
UR - https://arxiv.org/abs/2609.06755
AB - We prove that if H and K are subgroups of a free group F of arbitrary rank and H'=K'≠ 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.
ER -