On the non-existence of finite groups with certain normal subgroups

Ihechukwu Chinyere

On the non-existence of finite groups with certain normal subgroups

Group Theory 2026-01-08 v2

Authors: Ihechukwu Chinyere

Abstract

Problem 20.21 of Mazurov and Khukhro (Unsolved Problems in Group Theory: The Kourovka Notebook, 20th Issue, 2022), contributed by M.~Conder and attributed to G.~Verret, asks whether there exists a finite group $G$ with two normal subgroups $K$ and $L$ of index $12$ such that $K \cong L$ , but with non-isomorphic quotients $G/K \cong C_{12}$ and $G/L \cong A_4.$ We prove that no such finite group exists.

Keywords

finite group normal subgroup isomorphism index Kourovka Notebook

Cite

@article{arxiv.2601.01080,
  title  = {On the non-existence of finite groups with certain normal subgroups},
  author = {Ihechukwu Chinyere},
  journal= {arXiv preprint arXiv:2601.01080},
  year   = {2026}
}

Comments

We have discovered an error in Section 3.4 that affects the main arguments, and we wish to withdraw the paper until it can be corrected

On the non-existence of finite groups with certain normal subgroups

Abstract

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