From wikipedia and this question Why is the set of commutators not a subgroup?, I know that the set of commutators $G'=\{a^{-1}b^{-1}ab|a,b\in G\}$ of a group $G$ is not a group, because $G'$ is not closed. That is to say, the product of two commutators is not necessarily a commutator. Since $G'$ is not a group, it is not the subgroup of $G$. Then WHY $G'$ is called "commutator subgroup" or "derived subgroup"?
Group Theory – Why the Set of Commutators is Called Commutator Subgroup
group-theory
Best Answer
$G'$ is a subgroup of $G$. It can be defined in different ways. One way (which aptly explains and justifies its name) is that $G'$ is the smallest normal subgroup of $G$ such that $G/G'$ is commutative. Thus, $G'$, the commutator subgroup, is the smallest part of $G$ that needs to be killed in order to turn $G$ into a commutative group. Hence $G'$ 'commutates' $G$. (Proving this subgroup exists, without using any commutators, is a nice little exercise).
This same subgroup $G'$ can be defined as the subgroup of $G$ generated by all the commutators $[x,y]=xyx^{-1}y^{-1}$. Notice that groups exist (though it's not easy to find an example) where the set of all commutators is strictly smaller than the commutator subgroup. So, to answer your question, the set of all commutators is generally not called the commutator subgroup.