In [6], J. Tits has shown that the Ree group 2F4(2) is not simple but possesses a simple subgroup of index 2. In this paper we prove the following theorem:THEOREM. Let G be a finite group of even order and let z be an involution contained in G. Suppose H = CG(z) has the following properties:(i) J = O2(H) has order 29and is of class at least 3.(ii) H/J is isomorphic to the Frobenius group of order 20.(iii) If P is a Sylow 5-subgroup of H, then Cj(P) ⊆ Z(J). Then G = H • O(G) or G ≊ , the simple group of Tits, as defined in [6].For the remainder of the paper, G will denote a finite group which satisfies the hypotheses of the theorem as well as G ≠ H • O(G). Thus Glauberman's theorem [1] can be applied to G and we have that 〈z〉 is not weakly closed in H (with respect to G). The other notation is standard (see [2], for example).
Parrott, David. A Characterization of the Tits' Simple Group. Canadian journal of mathematics, Tome 24 (1972) no. 4, pp. 672-685. doi: 10.4153/CJM-1972-063-0
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title = {A {Characterization} of the {Tits'} {Simple} {Group}},
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