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On some extended dual polar spaces. I. (English) Zbl 0792.51009
An extended dual polar space is a polar space which is extended by a Buekenhout-\(c\)-geometry “at the side of the generalized quadrangle”, i.e. the \(c\)-stroke is next to the double stroke in the diagram. When classifying finite flag-transitive extended dual polar spaces, one can use the classification of the finite classical flag-transitive extended quadrangles (classical means that the generalized quadrangles in the residues are classical).
The paper under review deals with the determination of the flag- transitive simply connected extended dual polar spaces of rank 4 for which the corresponding generalized quadrangle is isomorphic to \(Q^ - _ 5(2)\) respectively \(W(2)\). In the latter case, the author has an extra assumption on the stabilizer of a flag of type \(\{0,3\}\) (i.e. flags the residue of which is the generalized quadrangle \(W(2))\). In each case, exactly 2 examples arise, amongst them one admitting a quotient having as full automorphism group Conway’s simple group \(Co_ 2\). The proof involves computations with commutation relations, regularly culminating in a coset enumeration.

MSC:
51E24 Buildings and the geometry of diagrams
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