Given a set S, the free group F(S) is the set of words using elements x of S and their formal inverses x⁻¹ with group operation being concatenation. Two distinct words are equal if and only if this involves replacing a product xx⁻¹ or x⁻¹x with the identity e or vice versa, that is, there are no non-trivial relations. Every group is a quotient group of a free group. See presentation.