Is a normal subgroup normal in a normal group?

No, take any group $G$ with a non-normal subgroup $H$ and equip $G$ with the discrete topology.


Not necessarily. Take any group $G$ endowed with the discrete metric and any non-normal (in the group-theoretical sense) subgroup $H$ of $G$. Then $H$ is normal with respect to the induced topology, but it is still a non-normal subgroup of $G$.

Tags:

General Topology

Group Theory

Topological Groups

Normal Subgroups

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