## For n at least 5, the commutator subgroup of Sym(n) is Alt(n) (via Project Crazy Project)

Prove that the commutator subgroup of $S_n$ is $A_n$ for all $n geq 5$. We know that $[S_n,S_n]$ is characteristic, hence normal, in $S_n$. Since $n geq 5$, then, this commutator is either 1, $A_n$, or $S_n$. Now $[S_n,S_n] neq 1$ since $S_n$ is nonabelian. Moreover, since $S_n/A_n cong Z_2$ is abelian, we have that $[S_n, S_n] leq A_n$. Thus $[S_n,S_n] = A_n$. … Read More