On the second-largest Sylow subgroup of a finite simple group of lie type

Let be a finite simple group of Lie type in characteristic , and let be a Sylow subgroup of with maximal order. It is well known that is a Sylow -subgroup except for an explicit list of exceptions and that is always 'large' in the sense that <![CDATA[|T|^{1/3}. One might anticipate that, moreover, the Sylow -subgroups of with are usually significantly smaller than . We verify this hypothesis by proving that, for every and every prime divisor of with , the order of the Sylow -subgroup of is at most , where is the Lie rank of.