I know that the statement "the union of two normal subgroups is also a normal subgroup" is false.
Is there a counterexample to show this?
I can prove that the intersection is normal, but I can't disprove this.
2026-04-01 17:35:51.1775064951
Prove that the union of two normal subgroups is also a normal subgroup
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$3\mathbb{Z}$ and $2\mathbb{Z}$ are subgroups of $(\mathbb{Z}, +)$ but the union has elements $2$ and $3$. But $2 + 3 = 5 \notin 3\mathbb{Z} \cup 2\mathbb{Z}$. $(\mathbb{Z}, +)$ is cyclic, and hence abelian, and hence all subgroups of $(\mathbb{Z}, +)$ are normal. But the union of these 2 subgroups is not even a subgroup.