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

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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 in an explicit list of exceptions, and that S is always `large' in the sense that |T|¹/³ < |S| |T|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 |T| with r ≠ p, the order of the Sylow r-subgroup of T at most |T|2ᵣ(4(ℓ+1) r)/ℓ=|T| O(ᵣ(ℓ)/ℓ), where ℓ is the Lie rank of T.