PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 129, Number 9, Pages 2571–2579 S 0002-9939(01)05940-8 Article electronically published on February 9, 2001 A REMARK ON A THEOREM OF J. TITS CURTIS D. BENNETT AND SERGEY SHPECTOROV (Communicated by Stephen D. Smith) Abstract. Let G be a rank two Chevalley group and Γ be the corresponding Moufang polygon. J. Tits proved that G is the universal completion of the amalgam formed by three subgroups of G: the stabilizer P 1 of a point a of Γ, the stabilizer P 2 of a line ℓ incident with a, and the stabilizer N of an apartment A passing through a and ℓ. We prove a slightly stronger result, in which the exact structure of N is not required. Our result can be used in conjunction with the “weak BN -pair” theorem of Delgado and Stellmacher in order to identify subgroups of finite groups generated by minimal parabolics. 1. Introduction In the continuing efforts to put together a complete and “simpler” proof of the classification of finite simple groups the problem of identifying the known simple groups from certain of their subgroups remains important. In particular, in one of the cases (the characteristic-p-type case) a simple Lie type group G in characteristic p must be reconstructed from its p-local subgroups. In the case where the rank of G is at least 3 there seems to be a clear “best” way to do this. Namely, G is known to be the universal closure of the amalgam of its maximal parabolic subgroups. There exists an easy geometric argument for this important fact, based on the simple connectedness of the building on which G acts (cf. Tits [T]). When the rank of G is two, the universal closure of the amalgam of the maximal parabolics, {P 1 ,P 2 }, is the free amalgamated product of P 1 and P 2 , and hence infinite. The day is saved by a theorem of Tits ([S], Chapter II, Theorem 8) stating that G is the universal closure of the amalgam of P 1 , P 2 and N , where N is the normalizer of a Cartan subgroup C from B := P 1 ∩ P 2 . (This definition does not work when the field is small. A better definition (see below) that works in all cases can be given in terms of the generalized quadrangle on which G acts.) The first proof of this result used the BN -pair technique; more recently an elementary proof was found based again on the simple connectedness of the rank three geometry of points, lines, and apartments of the generalized n-gon on which G acts. Suppose we have an injection φ of the amalgam {P 1 ,P 2 } into a group H . (We will use the bar notation for the images under this injection.) This amounts to finding in H a pair of subgroups { ¯ P 1 , ¯ P 2 } forming an amalgam isomorphic to {P 1 ,P 2 }. In many cases the subgroups ¯ P 1 and ¯ P 2 are readily available due to the classification by Delgado and Stellmacher [DGS] of “weak BN -pairs”. It seems desirable to Received by the editors January 24, 2000. 1991 Mathematics Subject Classification. Primary 20E42, 20D06, 51E12, 51E24. The second author received partial support from NSF grant #9896154. c 2001 American Mathematical Society 2571 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use