Is there any result about maximal abelian p-subgroups of the exceptional group E\_6(q), where q=p^a is prime power?



abelian p- subgroups of E_6(q)
Aspects of this question have been thoroughly treated in *The Classification of Finite Simple Groups*, Number 3, by Gorenstein-Lyons-Solomon, AMS, 1994: see especially their Table 3.3.1 for the Chevalley groups. In your notation (which differs somewhat from theirs), the $p$-rank is $16a$. This gives only the rank of a maximal *elementary* abelian $p$-subgroup, however.


P.S. Concerning maximal abelian $p$-subgroups of $E\_6$, the relevant table in Vdovin's thesis (linked by Nick Gill) seems to give the same answer $p^{16a}$. 
Probably the point here is that the 16 "commuting" positive roots yield the only possible maximal abelian $p$-subgroups in a Chevalley group, automatically elementary abelian because of the structure of root groups. The emphasis on $p$-rank comes mainly from the connection with cohomological support varieties and such. Of course, Sylow $p$-subgroups are all conjugate, so their subgroup structure is what one needs to know.


