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Abstract
We have seen a few groups, such as the symmetric group S3, the dihedral groups Dn, and the group M of rigid motions of the plane, in which one can compute easily using a list of generators and a list of relations for manipulating them. The rest of this chapter is devoted to the formal background for such methods. In this section, we consider groups which have a set of generators satisfying no relations other than ones [such as x(xy) = (xy)z] which are implied by the group axioms. A set S of elements of a group which satisfy no relations except those implied by the axioms is called free, and a group which has a free set of generators is called a free group. We will now describe the free groups.