Describe the Lower Central Series for $S_5$.
I know that the commutator subgroup $[G,G] = A_5$. So the first two terms of the series are $$ S_5 \supset A_5. $$
The next element is $[S_5,A_5].$
How do I find this?
I think that it can't be $\{e\}$ since that would say that $A_5$ is abelian. (maybe?)
Is $[S_5,A_5]$ necessarily normal in $A_5$? If it is, then $[S_5,A_5] = A_5,$ but I couldn't find that result. . .