Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 2 / 18G / c

A group is solvable when its derived serieseventually reaches the trivial group; equivalently, it has a subnormal series with abelian quotients.

The symmetric group $S_4$ is solvable becausehas abelian quotients $V_4$, $C_3$, and $C_2$, where $V_4$ is the Klein four-group. Thus

For $n\ge 2$, the commutator subgroup of $S_n$ is $A_n$. The group $A_5$ is nonabelian and simple, so its commutator subgroup is a nontrivial normal subgroup and must equal $A_5$ itself. Hence the derived series of $S_5$ isand never reaches the identity. Therefore