FREE GROUPS

Let x be a generating subset of a group G. Certain products of members of X and their inverses will be 1 whatever X and G are; for instance, xyyz−1zy−1y−1x−1. Other products, such as xyz or xx, will be 1 for some choices of X and G but not for other choices. Those pairs G and X for which a product of elements in X ∪ X−1 is 1 only when the properties holding in all groups require it to be 1 are obviously of interest.

They are called free groups; a more formal definition will be given later. If G is such a group, any function f from x to a group H can be extended uniquely to a homomorphism from G to H. For any g ∈ G can be written as xi1ε1 … xin εn where εt = ±1 and xir ε X for r = 1, …, n. Now suppose that g can also be written as xj1δ1 … xjm, δm where δs = ±1 and xjs ε X for s − 1, …, m. Then

and our assumption on G and X then tells us we must have

Hence the element of H given by (xi1 ƒ)ε1 … (xin ƒ)εn depends only on g and not on how g is written as a product of elements of X ∪ X−1. I t follows that we can define a function φ:G → H by requiring gφ to be this element. It is easy to check that φ is a homomorphism and that xφ − xƒ for all x ε X.