TITLE: Unsolvability of $S_{n}$
QUESTION [5 upvotes]: Is there a short proof for unsolvability of $S_{n}$ without the standard approach of proving the simplicity of $A_{n}$ ? This is good, however, and one can prove this only with basic group theory, without conjugacy classes ; I was just wondering if there was a shorter proof.

REPLY [13 votes]: This follows from the study of the derived (=commutator) subgroups: $[S_n,S_n]=A_n$, $[A_n,A_n]=A_n$ for all $n\ge5$. This is easier to prove than simplicity of $A_n$.