That group $PSL(n,q)=SL(n,q)/(center)$ is simple is proved on the following lines:
(1) $SL(n,q)$ acts on (non-zero elements of) $n$-dimensional vector space over $\mathbb{F}_q$.
(2) Action of $SL(n,q)$ is doubly transitive, hence for any $v\neq 0$, the $Stab(v)$ is maximal.
(3) $Stab(v)$ contains an abelian normal subgroup $K$, s.t. conjugates of $K$ generate $SL(n,q)$.
(4) Thus $SL(n,q)$ modulo kernel of action (which becomes equal to center) is simple.
This proof can be found in Suzuki's or Alperin-Bells book on Group Theory.
However, the arguments follow a very general theorem, known as Iwasawa theorem (1941). But the simplicity of $PSL(n,q)$ is know much before this theorem proved.
Question: How it was proved, before Iwasawa's theorem, that $PSL(n,q)$ is simple? [one may just outline the proof, and/or suggest references for it.]
Note: $PSL(n,q)$ is simple except $(n,q)=(2,2), (2,3)$. We do not consider it above.