Moufang set

In mathematics, a Moufang set is a particular kind of combinatorial system named after Ruth Moufang.
Definition.
A Moufang set is a pair formula_1 where "X" is a set and formula_2 is a family of subgroups of the symmetric group formula_3 indexed by the elements of "X". The system satisfies the conditions
Examples.
Let "K" be a field and "X" the projective line P1("K") over "K". Let "U""x" be the stabiliser of each point "x" in the group PSL2("K"). The Moufang set determines "K" up to isomorphism or anti-isomorphism: an application of Hua's identity.
A quadratic Jordan division algebra gives rise to a Moufang set structure. If "U" is the quadratic map on the unital algebra "J", let τ denote the permutation of the additive group ("J",+) defined by 
Then τ defines a Moufang set structure on "J". The Hua maps "h""a" of the Moufang structure are just the quadratic "U""a" . Note that the link is more natural in terms of J-structures.