A 2-local Characterization of M12

A characterization of the Mathieu group M12 is established; the characterization is used by Aschbacher and Smith in their classification of the quasithin finite simple groups. This is one in a series of papers providing simplified, modern, computer free treatments of the existence and uniqueness of the sporadic groups, and of the normalizers of subgroups of prime order and Sylow subgroups in these groups. Such information is the minimum necessary for purposes of the Classification of Finite Simple Groups. The series also seeks to avoid appeals to references other than basic texts like [A1] and [A2], or other papers in the series which operate under the same constraints. Here we treat the Mathieu group M12. To identify M12 in the context of the Classification, one needs a 2-local characterization of M12. The characterization given here is one chosen for the purposes of [AS], where the quasithin groups of even characteristic are classified. This is the place in the Classification where M12 arises. Here is our characterization: Theorem 1. Let G be a finite group, z an involution in G, H = CG(z), Q = O2(H), and X ∈ Syl3(H). Assume (a) Q is extraspecial of order 32, (b) H/Q ∼= S3 and CQ(X) = 〈z〉, and (c) z is not weakly closed in Q with respect to G. Then one of the following holds: (1) There is a normal E8-subgroup V of G with G/V ∼= L3(2). (2) G ∼= A8 or A9, and the two Q8-subgroups of Q are not normal in H. (3) G ∼= M12, and the two Q8-subgroups of Q are normal in H. Received August 16, 2002. 2000 Mathematics Subject Classification. 20D05. This work was partially supported by NSF DMS-0203417. c ©2003 University of Illinois