A group or other algebraic object is called non-Abelian if the law of commutativity does not always hold, i.e., if the object is not Abelian. For example, the group of nonsingular matrices is non-Abelian, as can be seen by comparing [1 | 0 0 | -1][0 | 1 -1 | 0] = [0 | 1 1 | 0] and [0 | 1 -1 | 0][1 | 0 0 | -1] = [0 | -1 -1 | 0].

Abelian | Abelianization | group | non-Abelian group | ring

Niels Henrik Abel