A note on groups with a BN-pair of spherical type

Abstract.

In this note we prove the uniqueness of U in a group G with a spherical split-BN-pair of rank $\geq 3$ ,i.e., if G has such a BN-pair with $B = U \cdot H, U$ a nilpotent normal subgroup of B, and $B_{G} = 1$, then $U = U^{+}$ and $G_0$ is a normal subgroup of G. Here $G_0$ is the corresponding group of Lie-type and $U^+$ the subgroup of $G_0$ generated by all root-subgroups corresponding to positive roots.