Maximal Orders of Abelian Subgroups in Finite Chevalley Groups

E. P. Vdovin

Matematičeskie zametki, Tome 69 (2001) no. 4, pp. 524-549

Résumé

In the present paper, for any finite group $G$ of Lie type (except for ${}^2F_4(q)$), the order $a(G)$ of its large Abelian subgroup is either found or estimated from above and from below (the latter is done for the groups $F_4(q)$, $E_6(q)$, $E_7(q)$, $E_8(q)$ and ${}^2E_6(q^2)$). In the groups for which the number $a(G)$ has been found exactly, any large Abelian subgroup coincides with a large unipotent or a large semisimple Abelian subgroup. For the groups $F_4(q)$, $E_6(q)$, $E_7(q)$, $E_8(q)$ and ${}^2E_6(q^2)$, it is shown that if an Abelian subgroup contains a noncentral semisimple element, then its order is less than the order of an Abelian unipotent group. Hence in these groups the large Abelian subgroups are unipotent, and in order to find the value of $a(G)$ for them, it is necessary to find the orders of the large unipotent Abelian subgroups. Thus it is proved that in a finite group of Lie type (except for ${}^2F_4(q)$) any large Abelian subgroup is either a large unipotent or a large semisimple Abelian subgroup.