I am working through an abstract algebra text and am stuck on a question. The problem statement is: A group $G$ is called perfect if $[G,G] = G$. Show that the alternating group $A_{n}$ is not perfect for $n \leq 4$ (without using the fact that it is solvable).
I think I'm misunderstanding the question. In the case that $n=3$ this makes sense to me because $A_{3} = \{e, (1 2 3), (1 3 2)\}$ but $[A_{3}, A_{3}] = \{e\}$ which is clearly not the entire group. But since $A_{1} = A_{2} = \{e\}$ and $eee^{-1}e^{-1} = e$ why are $A_{1}$ and $A_{2}$ not perfect?