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Rank one groups and division pairs

Rank one groups were introduced by F. G. Timmesfeld as the building
blocks of Lie type groups. Division pairs are algebraic objects categorically
equivalent to Moufang sets. We define a functor Delta from rank one groups to
division pairs and prove that Delta has a left adjoint Sigma, given by a
Steinberg type construction. As an application, we show that identities proved
by T. De Medts and Y. Segev for Moufang sets hold in arbitrary rank one groups.
We also extend the theory of the quasi-inverse, well known from Jordan pairs,
to this setting.
(Revised version 20 Jan 2014)

(This paper has appeard in Bull. Belg. Math. Soc. Simon Stevin **21**
(2014), 489--521)

O. Loos < ottmar.loos@fernuni-hagen.de >