I'm thinking that if $H\trianglelefteq G$, then $H$ is the only normal subgroup of $G$ of the order $|H|$, but I am not sure where to start or whether it's actually true or not.
2025-01-13 00:12:10.1736727130
Normal subgroup of index 2 is the only normal subgroup?
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It is not true. For example $\mathbb{Z}/2\mathbb{Z} \times \{0\}$ and $\{0\} \times \mathbb{Z}/2\mathbb{Z}$ are both normal subgroup of index 2 of $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$.