Simple Groups of Low 2-Rank

Abstract

With the completion of the classification of simple groups with quasidihedral or wreathed Sylow 2-subgroups (and with it of the 2-rank ≤ 2 theorem), a veritable “cottage industry” developed, having as its purpose the classification of simple groups having a Sylow 2-subgroup S isomorphic to that of some known simple group X (or family of simple groups). (For brevity, we say that S is of type X.) At first these were limited to cases in which S had 2-rank 3 or 4, but eventually led to groups or families in which S had higher rank—in some cases, even arbitrary rank. Indeed, there was a period in which it looked as though we were heading towards a characterization of every known simple group in terms of the structure of its Sylow 2-subgroups.