6
7
In Section 6 we sketch a proof that in a group G satisfying (G1), hypotheses (G2), (G2 ), and (G2 ) are equivalent.
There are existing characterizations of G 2 (3) in the literature which we will discuss in a moment. Our purpose here is to obtain a much shorter and simpler treatment for purposes of the classification, using modern methods which are more conceptual, avoid character theory, and minimize detailed computation. In the existing treatments, as in ours, the proof divides into two cases:
Case I: H is not strongly 3-embedded in G. Case II: H is strongly 3-embedded in G.
Thompson established the first characterization of G 2 (3) in terms of local information in the N-group paper [10]. His hypotheses involve restrictions on both 2-locals and 3-locals, and implicitly exclude Case II. The first characterization of G 2 (3) via the centralizer of an involution is due to Janko in [9]; he essentially assumes Hypotheses (G1) and (G2 ). In [8] and [7], Fong and Wong characterize groups with more general, but related centralizers; in the special case of G 2 (3) they appeal to Janko's paper to handle Case II. On the other hand Janko appeals to Thompson's work to handle Case I. Janko shows Case II leads to a contradiction using exceptional character theory. Both Fong-Wong and Thompson identify G as G 2 (3) in Case I by constructing a BN-pair for G.
We identify G in Case I: first by constructing a pair of 3-locals resembling the maximal parabolics in G 2 (3); then by appealing to work of Delgado and Stellmacher in [6] to conclude the amalgam determined by the 3-locals is unique up to isomorphism; and finally by an appeal to Corollary F.4.21 in [3] to identify G. In Case II we calculate the order of G by counting involutions, using an approach of Bender in [4]. This leads to an immediate contradiction via Sylow's Theorem.