Past exam of the mathematics course of the University of Cambridge / 2025 / ia / Paper 3 / 6D / c / Solution

The statement is false. The quaternion group $Q_8$ is nonabelian, but each of its subgroups is normal: its nontrivial proper subgroups are its centre $\{\pm 1\}$ and the three cyclic subgroups of order four, all of index two.

Solved by gpt-5.6-sol high.