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

TL;DR

This paper proves that no finite group exists with two normal subgroups of index 12 that are isomorphic but have non-isomorphic quotients, resolving a specific open problem in group theory.

Contribution

It establishes the non-existence of finite groups with two normal subgroups of index 12 that are isomorphic yet have different quotient structures.

Findings

No such finite group exists with the specified properties.

The problem posed in Mazurov and Khukhro's collection is resolved.

The result narrows the possibilities for the structure of finite groups with certain normal subgroups.

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.