A remark on a theorem of J. Tits (Q2716099)

Let \(\Gamma\) be a finite Moufang generalized \(n\)-gon, \(n\geq 3\) (or, equivalently, an irreducible spherical Moufang building of rank 2, i.e., the building arising from a finite Chevalley group of rank 2), and let \(G\) be an automorphism group of \(\Gamma\) containing the little projective group (hence containing all elations, or in other words, all root subgroups). Let \(\{x_1,x_2\}\) be a chamber of \(\Gamma\) and let \(P_i\) be the stabilizer of \(x_i\) in \(G\) (a so-called maximal parabolic), \(i=1,2\). Let \(N\) be the setwise stabilizer in \(G\) of an arbitrary but fixed apartment \(\Sigma\) containing \(x_1,x_2\), and let \(C\) be the pointwise stabilizer of \(\Sigma\) in \(G\). Further, set \(N_i:=N\cap P_i\), \(i=1,2\). Let \(H\) be an arbitrary group and suppose that \(\phi\) is an injective morphism of the amalgam \(\{P_1,P_2\}\) into \(H\). The result of Jacques Tits referred to in the title of the paper under review states that, if \(\phi\) extends to an injection of the amalgam \(\{P_1,P_2,N\}\) into \(H\), then \(\langle P_1^\phi,P_2^\phi\rangle\) is isomorphic to \(G\). The paper under review weakens the condition of that result. Namely, the authors show that the same conclusion is true if the group \(\langle N_1^\phi,N_2^\phi\rangle/C^\phi\), which is always a dihedral group, has finite order \(\leq 2n\) (in which case the order is automatically equal to \(2n\)). As a corollary, the authors show that the same conclusion is true if \(G\) is contained in the group generated by all root elations and all generalized homologies, and if \(C^\phi\) is self-centralized in \(H\).NEWLINENEWLINENEWLINEThe paper concludes with the cases where the corresponding rank 2 simple Chevalley group does not contain the little projective group. These are the cases (Atlas notation) \(S_4(2)\), \(G_2(2)\) and \(^2F_4(2)'\). The first two appear to be true exceptions for the Main Result, while for the third group the Main Result is valid, putting \(G={^2F_4(2)'}\) (and \(n=8\)).