Let $x$ and $y$ be two element of general linear group $SL(n,q)$, such that the orders of $x$ and $y$ be some primitive prime divisor of $q^n-1$. Is it true that if $xy\not=yx$, then $x$ and $y$ generate $SL(n,q)$ ? 


Note that the order of an element like $x$ of $SL(n,q)$ is a primitive prime divisor of $q^n-1$ if and only if $|x|\mid (q^n-1)$ but $|x|\not\mid(q^k-1)$ for every $k<n$.



The generators of special linear groups
Not necessarily. For $n$ even, they could generate one of the smaller classical groups ${\rm Sp}(n,q)$ or $\Omega^-(n,q)$.


For $n$ odd, it might be true sometimes - I think it is true in ${\rm SL}(5,2)$ for example. But there will sometimes be exceptions. For example, in ${\rm SL}(5,3)$ $x,y$ could generate $L\_2(11)$ or $M\_{11}$.


