Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 44 (2003), No. 1, 213-227.

Groups with Root-System of Type BC_{ℓ} Franz Georg Timmesfeld Mathematisches Institut, Justus-Liebig-Universität Gießen Arndtstraße 2, D-35392 Gießen, Germany 1. Introduction Let B be an irreducible, spherical Moufang building of rank ℓ ≥ 2, A an apartment of B and Φ the set of roots (half-apartments) of A with corresponding root-subgroups A_{r}, r ∈ Φ in the sense of Tits. Then we call G = ⟨A_{r} | r ∈ Φ⟩ ≤ Aut(B) the group of Lie-type B. The notion of a group of Lie-type B is very general, since it includes:

- simple classical groups over division rings of finite Witt-index ℓ ≥ 2,

- simple algebraic groups over arbitrary fields of relative rank ℓ ≥ 2,

- the finite simple groups of Lie-type of rank ℓ ≥ 2. The theory of such groups of Lie-type B was developed in [8], see also [3, I §4 and II §5]. In particular it was shown that one can enlarge Φ to some possibly nonreduced root-system Φ̃ (Φ ≠ Φ̃ only if Φ is of type B_{ℓ} and Φ̃ of type BC_{ℓ} or Φ is of type I_{2}(8) and Φ̃ of type ^{2}F_{4}; for the latter see [9, (5.4)]) such that the A_{r}, r ∈ Φ̃, satisfy:

(1) X_{r} = ⟨A_{r}, A_{−r}⟩ is a rank one group with unipotent subgroups A_{r} and A_{−r} for r ∈ Φ̃. (For definition of a rank one group see [3, I].). Further A_{2r} ≤ A_{r} if also 2r ∈ Φ̃.

(2) If r, s ∈ Φ̃ with s ≠ −r and −2r, then

[A_{r}, A_{s}] ≤ ⟨A_{λr+μs} | λr + μs ∈ Φ̃ and λ, μ ∈ ℕ⟩

(We use the convention ⟨∅⟩ = 1. Hence (2) implies A′_{r} = 1 if 2r ∉ Φ̃ and A′_{r} ≤ A_{2r} ≤ Z(A_{r}) if 2r ∈ Φ̃!)

Now it would be desirable to prove the converse. That is to show that, if G is a group generated by nonidentity subgroups A_{r}, r ∈ Φ̃ and Φ̃ as above, satisfying (1) and (2), then

0138-4821/93 $ 2.50 \begin{matrix}c \\ ⊗ \end{matrix} 2003 Heldermann Verlag