In a note I saw this fact that $PSL(3,q)$ where $q=p^n$ does not have any abelian subgroup of order $q^3$. But I could not prove it or find any reference about it, could you please help me about it? Thanks
One of the Sylow $p$-subgroups of $PSL(3,q)$ is the subgroup of all unitriangular matrices (i.e. upper triangular matrices with 1 on the diagonal): just compute the order of $PSL(3,q)$ and the order of the unitriangular subgroup which is $q^3$ (or look in Bogopolsky's group theory book). Therefore if $PSL(3,q)$ contained an Abelian subgroup of order $q^3$, the Sylow $p$-subgroup would be Abelian, but it is not (it is a finite version of the Heisenberg group).