A group is called a free group if no relation exists between its group generators other than the relationship between an element and its inverse required as one of the defining properties of a group.
For example, the additive group of integers is free with a single generator, namely 1 and its inverse, . An example of an element of the free group on two generators is , which is not equal to . The fundamental group of the figure eight serves as another good example of a free group with two generators, since either loop can be traversed, but the two paths do not commute. Moreover, no (nontrivial) path involving more than one loop will ever be homotopic to the identity.