operation . Every Steiner triple system defines an idempotent, commutative quasigroup: is the third element of the triple containing a and b. These quasigroups also satisfy for all x and y in the quasigroup. These quasigroups are known as Steiner quasigroups. The set where and with all other products as in the quaternion group forms a nonassociative loop of order 8. See hyperbolic quaternions for its application. (The hyperbolic quaternions themselves do not form a loop or quasigroup.) The nonzero octonions form a nonassociative loop under multiplication. The octonions are a special type of loop known as a Moufang loop. An associative quasigroup is either empty or is a group, since if there is at least one element, the invertibility of the quasigroup binary operation combined with associativity implies the existence of an identity element which then implies the existence of inverse elements, thus satisfying all three requirements of a group. The following construction is due to Hans Zassenhaus. On the underlying set of the four-dimensional vector space F4 over the 3-element Galois field define (x1, x2, x3, x4) ∗ (y1, y2, y3, y4) = (x1, x2, x3, x4) + (y1, y2, y3, y4) + (0, 0, 0, (x3 − y3)(x1y2 − x2y1)). Then, is a commutative Moufang loop that is not a group. More generally, the nonzero elements of any division algebra form a quasigroup. Properties In the remainder of the article we shall denote quasigroup multiplication simply by juxtaposition. Quasigroups have the cancellation property: if , then . This follows from the uniqueness of left division of ab or ac by a. Similarly, if , then . The Latin square property of quasigroups implies that, given any two of the three variables in , the third variable is uniquely determined. Multiplication operators The definition of a quasigroup can be treated as conditions on the left and right multiplication operators , defined by The definition says that both mappings are bijections from Q to itself. A magma Q is a quasigroup precisely when all these operators, for every x in Q, are bijective. The inverse mappings are left and right division, that is, In this notation the identities among the quasigroup's multiplication and division operations (stated in the section on universal algebra) are where 1 denotes the identity mapping on Q. Latin squares The multiplication table of a finite quasigroup is a Latin square: an table filled with n