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 $$\left({ X; \{U_x\}_{x \in X} }\right)$$ where X is a set and $$\{U_x\}_{x \in X}$$ is a family of subgroups of the symmetric group $$\Sigma_X$$ indexed by the elements of X. The system satisfies the conditions
 * $$U_y$$ fixes y and is simply transitive on $$X \setminus \{y\}$$;
 * Each $$U_y$$ normalises the family $$\{U_x\}_{x \in X}$$.

Examples
Let K be a field and X the projective line P1(K) over K. Let Ux 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
 * $$ x \mapsto -x^{-1} = - U_x^{-1}(x) \ . $$

Then τ defines a Moufang set structure on J. The Hua maps ha of the Moufang structure are just the quadratic Ua. Note that the link is more natural in terms of J-structures.