= Perfectness of the special linear group
For dimension at least three, $[I+aE_{ik},I+E_{kj}]=I+aE_{ij}$ 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 $s^2\ne1$ and commute an elementary matrix with $\operatorname{diag}(s,s^{-1})$. In both cases all generating <transvections> lie in the <commutator subgroup>, so the <special linear group over a finite field> is perfect outside $\operatorname{SL}_2(\mathbb F_2)$ and $\operatorname{SL}_2(\mathbb F_3)$.
// Target: algebra.bigb
Back to article page
