In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure that resembles a group in the sense that "division" is always possible. Quasigroups differ from groups mainly in that the associative and identity element properties are optional. In fact, a nonempty associative quasigroup is a group.

A quasigroup that has an identity element is called a loop.

There are at least two structurally equivalent formal definitions of a quasigroup:

The homomorphic image of a quasigroup that is defined with a single binary operation, however, need not be a quasigroup, in contrast to a quasigroup as having three primitive operations. We begin with the first definition.

A quasigroup (Q, ∗) is a set Q with a binary operation ∗ (that is, a magma, indicating that a quasigroup has to satisfy the closure property), obeying the Latin square property. This states that, for each a and b in Q, there exist unique elements x and y in Q such that both $${\displaystyle a\ast x=b}$$ $${\displaystyle y\ast a=b}$$ hold. (In other words: Each element of the set occurs exactly once in each row and exactly once in each column of the quasigroup's multiplication table, or Cayley table. This property ensures that the Cayley table of a finite quasigroup, and, in particular, a finite group, is a Latin square.) The requirement that x and y be unique can be replaced by the requirement that the magma be cancellative.

The unique solutions to these equations are written x = a \ b and y = b / a. The operations '\' and '/' are called, respectively, left division and right division. With regard to the Cayley table, the first equation (left division) means that the b entry in the a row is in the x column while the second equation (right division) means that the b entry in the a column is in the y row.

The empty set equipped with the empty binary operation satisfies this definition of a quasigroup. Some authors accept the empty quasigroup, but others explicitly exclude it.

Given some algebraic structure, an identity is an equation in which all variables are tacitly universally quantified, and in which all operations are among the primitive operations proper to the structure. Algebraic structures that satisfy axioms that are given solely by identities are called varieties. Many standard results in universal algebra hold only for varieties. Quasigroups form a variety if left and right division are taken as primitive.