ON THE SECOND-LARGEST SYLOW SUBGROUP OF A FINITE SIMPLE GROUP OF LIE TYPE

Let T be a finite simple group of Lie type in characteristic p, and let S be a Sylow subgroup of T with maximal order. It is well known that S is a Sylow p-subgroup except for an explicit list of exceptions and that S is always 'large' in the sense that vertical bar T vertical bar(1/3) < vertical bar S vertical bar <= vertical bar T vertical bar(1/2). One might anticipate that, moreover, the Sylow r-subgroups of T with r # p are usually significantly smaller than S. We verify this hypothesis by proving that, for every T and every prime divisor r of vertical bar T vertical bar with r not equal p, the order of the Sylow r-subgroup of T is at most vertical bar T vertical bar(2 left pependicular log)(r(4(l+1)r))( Right pependicular)( / l) = vertical bar T vertical bar(O(logr)((l)/l)), where Pis the Lie rank of T.