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.
