Sylow 2-Subgroups of Finite Simple Groups

Abstract

To one’s surprise, it was only until the late of the 19th century that a mathematician announced the classification of all groups of order 12. Unfortunately there was an error. Three years later, in 1899, Cayley showed it correctly. Namely, there are five nonisomorphic groups of order 12. One hundred years is long enough for mathematicians to make a quantum leap, since in the year 2000, Besche, Eick, and O’Brien [BEO] determined all isomorphism classes of groups of order ≤ 2000. Among them, there are exactly 49,487,365,422 groups of order 1024 = 2¹⁰. All others count 423,164,062 in number. In other words, 99.16% of all groups of order ≤ 2000 are of just one order 2¹⁰. (If we add groups of order 512, 128, etc., the ratio will be only a little greater for 2-groups.) Asymptotically perhaps:

Almost all finite groups are 2-groups.