A Characterization of Chevalley Groups Over Fields of Odd Order

THEOREM I. Let G be a finite group with F*(G) simple. Let z be an involution in G and K a subnormal subgroup of CQ(z) such that K has nonabelian Sylow 2-subgroups and z is the unique involution in K. Assume for each 2-element k C K that kG n C(z) C N(K) and for each g e C(z) N(K), that [K, Kg]<! O(C(z)). Then F*(G) is a Chevalley group of odd characteristic, M11, M12 or SP6(2). COROLLARY II. Let G be a finite group with F*(G) simple and let K be tightly embedded in G such that K has quaternion Sylow 2-subgroups. Then F*(G) is a Chevalley group of odd characteristic, M11, or M12. COROLLARY III. Let G be a finite group with F*(G) simple. Let z be an involution in G and K a 2-component or solvable 2-component of CQ(z) of 2-rank 1, containing z. Then F*(G) is a Chevalley group of odd characteristic or M,1. Theorem I follows from Theorems 1 through 8, stated in Section 2, which supply more specific information under more general hypotheses. Corollary II follows directly from Theorem I. Corollary III follows from Theorem I and Theorem 3 in [3]. All Chevalley groups with the exception of L2(q) and 2G2(q) satisfy the hypotheses of Theorem I. Terminology and notation are defined in Section 2. The possibility of such a theorem was first suggested by J.G. Thompson in January 1974, during his lectures at the winter meeting of the American Mathematical Society in San Francisco. At the same time Thompson also pointed out the significance of a certain section of the group, which is crucial to the proof. We have taken the liberty of referring to this section as the Thompson group of G; see Section 2 for its definition. The theorem finds its motivation in the study of component-type groups. Some applications to this theory are described in [6]. The remainder of this