A new proof of a theorem of Phan

Abstract

We apply diagram geometry and amalgam techniques to give a new proof of a theorem of K.-W. Phan, characterizing the special unitary group as a group generated by certain systems of subgroups SUð2; q 2 Þ.

On flips of unitary buildings I: Classification of flips

“…These so-called "Phan-type" theorems have been studied in a number of papers (e.g. [4], [3], [7]) initially in order to aid the Gorenstein-Lyons-Solomon revision of the proof of the Classification of Finite Simple Groups. Roughtly speaking, these "Phantype" theorems allow for the recognition of a group based on amalgams of subgroups that are produced by the group acting on a geometry.…”

On the hyperbolic unitary geometry

“…The long root subgroups of SU n (q 2 ) are abelian, conjugate in SU n (q 2 ) (as SU n (q 2 ) acts transitively on the set of isotropic one-dimensional subspaces of V ), and generate SU n (q 2 ) (see, e.g., [2]). Moreover, depending on whether two isotropic onedimensional subspaces a, b of V are perpendicular or not, the corresponding long root subgroups U a and U b commute or generate a (fundamental) SL 2 .…”

Developments in finite Phan theory

“…The first step has been well understood by now, cf. [21,54], also [100]. Therefore in this survey I will only concern myself with the second step.…”

“…In [21] this simple connectedness is shown by proving that every cycle of the flag complex of G A n is null-homotopic, while in [47] it is proved in odd characteristic by studying certain subgroup complexes of SU n+1 (q 2 ). …”

“…In this survey I will sketch the proof given in [21]. Fixing the base element x to be a point, a standard technique based on residual connectedness allows to reduce every cycle of G A n to a cycle in the point-line incidence graph, i.e., the graph on the elements of dimension one and two with incidence as adjacency.…”