I have been studying commutators and commutator subgroups, and I understand that commutators are elements of the form $xyx^{-1}y^{-1}$ for $x$ and $y$ in group $G$.
I know that the commutator subgroup is the smallest normal subgroup $N$ in $G$ for which $G/N$ is abelian.
I also know that the commutator subgroup is the subgroup generated by all the commutators. (I don't really understand the definition, especially in light of the facts that I am about to state below). It sounds like this definition is saying that the set of all commutator elements generates it's own subgroup...? How can multiple elements generate a group - are the commutators ordered pairs? So, this definition actually means to say that the commutator subgroup is the subgroup generated by an ordered pair containing all commutators?
The facts that confuse me (that I just said I would state "below") in trying to understand the definition are: why may commutator subgroups contain elements that are not commutators? and, why the set of all commutators is not itself the commutator subgroup?
Is a commutator subgroup a group generated from a single commutator, and therefore there are as many commutator subgroups as there are commutators?
I guess I am having trouble understanding what a commutator subgroup is, when its elements are not the commutators themselves. I have a textbook (which I did read many times) and I read multiple definitions from multiple institutions online, and read a definition in my old/different textbook, and watched a couple YouTube videos and read some other responses in this website and none seem to answer my question.
I even know/understand some proofs dealing with commutators (results like G/N will be abelian, and if the commutator is the whole group then $G$ is perfect, and some other concepts that I do understand) but these other proofs and resources don't seem to tell me what commutator subgroups actually are.
I don't even know what I am not understanding about this, when I know the definitions and i understand some results, so it makes me feel like I am missing "the point" somehow. Can anyone who understands what I am confused about please clarify for me what my question really is or what the commutator subgroup is?
