I need a little help to understand this answer.
Let $G$ be a group, and $N$ be the subgroup generated by elements of the form $aba^{-1}b^{-1}$, where $a,b \in G$. I've shown that this subgroup is normal in $G$ and that the quotient group $ G / N $ is abelian.
Problem: Find $N$ if $G = S_3 $.
Solution: We have that $A_3$ is a normal subgroup of index 2, and it follows that $N$ is a subgroup of $A_3$.
$N = A_3$.
My question: Why does it follow that $N$ is a subgroup of $A_3$? Is that always the case, for when we have a normal subgroup of index $2$, any other normal subgroup must be a subgroup of the normal subgroup? I am aware that one can calculate all the subgroups of $S_3$ and find that it's only nontrivial normal subgroup is $A_3$.
Best regards