A note on groups with a BN-pair of spherical type. (Q1881526)

In this note, the author shows that, if a group \(G\) has a BN-pair of rank at least 3 enjoying the property that there exists a nilpotent subgroup \(U\trianglelefteq B\) such that \(B=U\cdot(B\cap N)\), then \(G\) contains the corresponding group \(G_0\) of Lie type (uniquely determined by the spherical building defined by \(G\)), and \(U\) is the standard unipotent subgroup (usually denoted by \(U^+\)) generated by all root groups corresponding to positive roots. The method of the proof is by looking at the possible groups with a BN-pair of rank 2 that turn up, and prove the result for them via the different rank 1 groups. The same result extended to the rank 2 case is proved by \textit{T.~De Medts, F.~Haot, K.~Tent} and the reviewer [in J.~Group Theory 8, No. 1, 1-10 (2005; Zbl 1091.20023)], using a more geometric argument, requiring less work in the rank 1 case.