Vol. 10, No. 1, 2009

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Moufang sets related to polarities in exceptional Moufang quadrangles of type $\mathsf{F}_4$

Koen Struyve

Vol. 10 (2009), No. 1, 121–132

Departing from a Moufang set related to a polarity in an exceptional Moufang quadrangle of type F4, we construct a rank three geometry. The main property of this new geometry is that its automorphism group is identical to the one of the underlying Moufang set, providing a tool to study this Moufang set in a geometrical way. As a corollary we obtain that every automorphism of an exceptional Moufang quadrangle of type F4 stabilizing the absolute points of a polarity, also centralizes the polarity. This handles the final case of a similar result for all polarities of Moufang n-gons with n 3.

Moufang sets, exceptional quadrangles of type F_4, polarities
Mathematical Subject Classification 2000
Primary: 20B22, 51E05, 51E12
Received: 10 January 2008
Accepted: 30 May 2009
Koen Struyve