Perfect group

A group $G$ is perfect when $G^{\prime }=G$, equivalently when its abelianization is trivial.

For dimension at least three, $[I+a{E}_{ik},I+E_{kj}]=I+a{E}_{ij}$ for distinct $i,j,k$, with $[x,y]=x^{-1}{y}^{-1}xy$. In dimension two over a field with more than three elements, choose $s$ with $s^2\ne 1$ and commute an elementary matrix with $\mathrm{diag}(s,s^{-1})$. In both cases all generating transvections lie in the commutator subgroup, so the special linear group over a finite field is perfect outside ${\mathrm{SL}}_2({\mathbb{F}}_2)$ and ${\mathrm{SL}}_2({\mathbb{F}}_3)$.

// Target: algebra.bigb