Hello I know that there are abelian subgroups of order (q^2-1)/d (cyclic group) and (q-1)^2/d and q^2(elementary abelian) and (q-1)q/d in the group PSL(3,q), where d=(3,q-1). Also I saw in a text that

1) The orders of maximal abelian subgroups which divide (q-1)^2(q+1) are equal to (q-1)^2/d.

and 2) The orders of maximal abelian subgroups which divide q^2(q-1)^2 are equal to q(q-1)/d.

and 3) The orders of maximal abelian subgroups which divide q^3 are equal to q^2.

But I don't know the reason of the above results, could you kindly help me and let me know why the above facts are true?

Best regards Darya