Background

A free group on a set S is a group where each element can be uniquely described as a finite length product of the form:
: where the s_i are elements of S, adjacent s_i are distinct, and a_i are non-zero integers (but n may be zero). In less formal terms, the group consists of words in the generators and their inverses, subject only to canceling a generator with an adjacent occurrence of its inverse.

If G is any group, and S is a generating subset

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