Q:

any subgroup is a union of normal subgroups?

Let $G$ be an abstract group, and $A$ is a subgroup of $G$, then can $A$ be written as an union of $A_\alpha$, $\alpha\in I$, where $A_\alpha$ are normal subgroup of $G$?
Thanks!

A:

Let it be that $A$ is a subgroup of $G$ that can be written as a union of normal subgroups. 
Then for every $x\in A$ there is some normal subgroup $N_x$ such that $x\in N_x\subseteq A$.
Then for every $x\in A$ and every $g\in G$ we find that: $$gxg^{-1}\in N_x\subseteq A$$
So actually it is proved by this that $A$ is a normal subgroup.
Conclusion: only normal subgroups of $G$ can be written as a union of normal subgroups.

A:

Consider $G=S_3$.
Then $H=\{1,(12)\}$ is a subgroup which is not normal.
Clearly, $H$ cannot be written as a union of normal subgroups of $G$ as the only normal subgroups of $G$ are $1,G$ and $\{1,(123),(132)\}$.

