Projective special linear group over a finite field
= Projective special linear group over a finite field
{title2=$PSL_n(q)=SL_n(q)/Z(SL_n(q))$}
The scalar center has order $\gcd(n,q-1)$. For $n>1$ the quotient is simple except at $(n,q)=(2,2),(2,3)$. Its faithful primitive projective-point action, abelian fixed-center transvections, elementary-matrix generation and perfectness give the <Iwasawa simplicity lemma> proof.