Difficult# Perfect Groups: Definition , Identifying Perfect Groups

ABSALG-LFMBK0

A group $G$ is defined to be perfect if and only if $G=[G,G]$, where $[G,G]$ is the commutator or derived subgroup of $G$.

**Which of the following groups are perfect? **

The group $S_n$ is the symmetric group on $n$ symbols and $A_n$ is the alternating group, the subgroup of even permutations in $S_n$.