On the intersections of Sylow subgroups in almost simple groups

Let G be a finite almost simple group and let H be a Sylow p-subgroup of G. As a special case of a theorem of Zenkov, there exist x,y∈G such that H∩H^x∩H^y=1. In fact, if G is simple, then a theorem of Mazurov and Zenkov reveals that H∩Hx=1 for some x∈G. However, it is known that the latter property does not extend to all almost simple groups. For example, if G=S₈ and p=2, then H∩Hx≠1 for all x∈G. Further work of Zenkov in the 1990s shows that such examples are rare (for instance, there are no such examples if p⩾5) and he reduced the classification of all such pairs to the situation where p=2 and G is an almost simple group of Lie type defined over a finite field 𝔽q and either q=9 or q is a Mersenne or Fermat prime. In this paper, by adopting a probabilistic approach based on fixed point ratio estimates, we complete Zenkov's classification.