Free groups; generators and relations

Suppose a group G with neutral element 1 is generated by a set X ≠ 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315136547/a57ae1f0-6f59-46bd-bbdd-35a521b486a1/content/eq2758.tif"/> . What can we say about G? Well, certainly each element of G different from 1 can be written in the form x 1 ε 1 ⋯ x n ε n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315136547/a57ae1f0-6f59-46bd-bbdd-35a521b486a1/content/eq2759.tif"/> where n ∈ ℕ, x_i ∈ X and ε_i ∈ {±1}. Now not all of these product expressions represent different elements of G because there might be relations between the different generators. For example, if a, b, c are three different elements of X, equations like a ²= b ³ or b ⁴ = 1 might hold. In fact, there are relations which must hold according to the axioms of a group, namely xx ⁻¹= x ⁻¹ x = 1 for all x ∈ X; hence, for example, abb ⁻¹ c ⁻¹ ca ⁻¹ = 1 for all a, b, c ∈ X. If there are no relations between the elements of X other than the ones imposed by the group axioms we call the group G free on X (namely, free of non-mandatory relations). In this case, G will consist exactly of the neutral element 1 and all products x 1 ε 1 ⋯ x n ε n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315136547/a57ae1f0-6f59-46bd-bbdd-35a521b486a1/content/eq2760.tif"/> as above where two formally different products represent the same element of G if and only if one of the products can be obtained from the other by repeatedly cancelling or inserting terms of the form xx ⁻¹ or x ⁻¹ x where x ∈ X. (Of course, we use the abbreviations a ³≔ a · a · a and so on.) Let us give an example of a free group before we continue our general discussion.