A quasigroup is a grouplike structure $(Q, \ast)$, that satisfies the Latin square property but need not have an identity element, nor need it be associative. It coincides with the notion of a divisible magma.
A quasigroup is a grouplike structure $(Q, \ast)$, where $Q$ is a set and $$\ast: Q \times Q \to Q$$ is a binary product that satisfies the satisfying the Latin square property (or divisibility property), namely that for all $q, r \in Q$ there is a unique $a \in Q$ such that $r = a q$ and a unique $b \in Q$ such that $r = q b$, or equivalently, the multiplication table of $\ast$ is a Latin square.
A quasigroup $(Q, \ast)$ with an identity element (an element $1 \in Q$ such that $1q = q = q1$ for all $q \in Q$) is called a loop, and an associative loop is the same thing as a group.
Apart from groups, important examples of quasigroups include the integers $\mathbb{Z}$ endowed with subtraction $-$, $\mathbb{Q}$ endowed with division $\div$, vector spaces over fields of characteristic not $2$ endowed with the map $x \ast y := \frac{1}{2} (x + y)$. The nonzero elements of any division algebra always form a quasigroup; in the case of the octonions, the quasigroup of nonzero elements is a Moufang loop (a loop satisfying an additional algebraic identity) but not a group.
Quasi-group (Encyclopedia of Mathematics)
Quasigroup (Wikipedia)
J. Dénes, A.D. Keedwell, Latin squares and their applications, English Univ. Press (1974)
R. Moufang, Zur Struktur von Alternativkörpern Math. Ann., 110 (1935) pp. 416–430 (in German)
