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Sektion 13
Freitag, 22.09.2000, 17.30–17.50 Uhr, POT 13

Irreduzible BN-Paare von Rang 2

Katrin Tent, Universität Würzburg

Let G be a group with an irreducible spherical (B,N)-pair of rank 2 where B has a normal subgroup U with B = UT for T = B  /~\ N. Let P be the generalized n-gon associated to this (B,N)-pair and let W be the associated Weyl group. So T stabilizes an ordinary n-gon in P, and |W | = 2n. We prove that, if either U is nilpotent or G acts effectively on P and Z(U)/=1, then |W | = 2n with n = 3, 4, 6, 8 or 12. If G acts effectively and n/=4, 6, then (up to duality) Z(U) consists of central elations. Also, if n = 3 and U is nilpotent, then P is a Moufang projective plane and if, moreover, G acts effectively on P, then it contains its little projective group. Finally, we show that, if G acts effectively on P, if Z(U)/=1, and if T satisfies a certain strong transivity assumption, then P is a Moufang n-gon with n = 3, 4 or 6 and G contains its little projective group.

Under the model theoretic assumption that G has finite Morley rank and the BN-pair is definable stronger results can be obtained.