#### Vol. 10, No. 1, 2009

 Recent Issues Volume 18 Volume 16 Volume 15 Volume 14 Volume 13 Volume 12 Volume 11 Volume 10 Volume 9 Volume 8 Volume 6+7 Volume 5 Volume 4 Volume 3 Volume 2 Volume 1
 The Journal About the Journal Editorial Board Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN (electronic): 2640-7345 ISSN (print): 2640-7337 Author Index To Appear Other MSP Journals
Moufang sets related to polarities in exceptional Moufang quadrangles of type $\mathsf{F}_4$

### Koen Struyve

Vol. 10 (2009), No. 1, 121–132
##### Abstract

Departing from a Moufang set related to a polarity in an exceptional Moufang quadrangle of type ${\mathbb{F}}_{4}$, 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 ${\mathbb{F}}_{4}$ 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 $n\ge 3$.

##### Keywords
Moufang sets, exceptional quadrangles of type F_4, polarities
##### Mathematical Subject Classification 2000
Primary: 20B22, 51E05, 51E12