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1.9. FREE GROUPS, GENERATORS AND RELATIONS, AND FREE PRODUCTS 29

1.9 Free Groups, Generators and Relations, and Free Products

We shall show that free objects (free groups) exist in the , and we shall use these to develop a method of describing groups in terms of “generators and relations.” In addition, we indicate how to construct (free products) in the category of groups.

1.9.1 Free Groups We shall construct a F (X) that is free on a given set X. If X = ∅ then the group free on X is F (X) = {e}. Choose a set X−1 disjoint from X and satisfying that |X−1| = |X|. Fix a bijection X → X−1. The of x ∈ X in X−1 is denoted x−1. Choose a singleton set disjoint −1 from X ∪ X and denote its element 1. A on X is a sequence (a1, a2, ··· ) with −1 + ai ∈ X ∪ X ∪ {1} and for some n ∈ Z , ak = 1 for k ≥ n. Use 1 to denote the sequence (1, 1, ··· ). A word is said to be reduced if

1. for all x ∈ X, x and x−1 are not adjacent;

2. ak = 1 implies ai = 1 for all i ≥ k.

λ1 λ2 λn λ1 λ2 λn −1 Write a reduce word (x1 , x2 , ··· , xn , 1, 1, ··· ) by x1 x2 ··· xn , where xi ∈ X ∪ X and λi ∈ {±1}. Let F := F (X) denote the set of all reduced words on X. Define a binary operation (called ) on F (X) by juxtaposition and cancelation of adjacent terms of the form xx−1 or −1 x x. For example, if xi are distinct for i = 1, ··· , 5, then

−1 −1 −1 −1 (x1x2x3 x4)(x4 x3x5x5x6 ) = x1x2x5x5x6 . Thm 1.44. F (X) defined above is a group and F (X) = hXi.

F (X) is called the on the set X.

Thm 1.45 (F (X) is a on X in the category of groups). Let X be a nonempty set. Let ι : X → F (X) be the . If G is a group and f : X → G a map of sets, then there exists a unique f¯ : F (X) → G such that f¯◦ ι = f.

F (X) O f¯ ι " X / G f

Cor 1.46. Every group G is the homomorphic image (i.e. the image of an ) of a free group. 30 CHAPTER 1. GROUPS

Proof. Let X be a set of generators of G. By Theorem 1.45, the inclusion map X → G induces a homomorphism f¯ : F (X) → G such that x 7→ x ∈ G. Since G = hXi, the map f¯ is an epimorphism.

1.9.2 Generators and Relations In Corollary 1.46, G ' F (X)/N where N := Ker f¯ is a normal of F (X). It suggests that every group G can be determined by a set of generators (=X), and a set of relations (= a set of generators of N). f¯ We use the same symbol x to denote the images of x ∈ X in the mapping X →ι F (X) → G. δ1 δn δ1 δn Let w = x1 ··· xn be a generator of N. Under the epimorphism F (X) → G, w 7→ x1 ··· xn = δ1 δn e ∈ G. The equation x1 ··· xn = e in G is called a relation on the generators xi. Conversely, given a set X and a set Y of reduced words on X, there is a group G generated by X and all the relations w = e (w ∈ Y ).1 Let us construct G. First, construct a free group F (X) on X. Second, find the N(Y ) of F (X) generated by Y .2 Then G ' F (X)/N(Y ) is the desired group. The group G ' F (X)/N(Y ) is called the group defined by the generators x ∈ X and relations w = e (w ∈ Y ). One says that (X|Y ) is a presentation of G.

Thm 1.47. Let X be a set, Y a set of words on X, G the group defined by the generators x ∈ X and relations w = e (w ∈ Y ). If H is any group such that H = hXi and H satisfies w = e (w ∈ Y ), then there is an epimorphism G → H.

Proof. Let F (X) be a free group on X. The inclusion map X → H induces an epimorphism ϕ : F (X) → H. Let N(Y ) be the normal subgroup of F (X) generated by Y . Then Ker ϕ ≥ N(Y ). There is an induced group epimorphism G ' F (X)/N(Y ) → F (X)/Ker ϕ ' H.

Ex. Let G be the group defined by generators a, b and relations a4 = e, a2b−2 = e and −1 abab = e. On one hand, the Q8 of 8, is generated by a, b satisfying these relations. So there is an epimorphism ϕ : G → Q8. Then |G| ≥ |Q8|. On the other hand, ba = a−1b and b2 = a2 in G. So every element of G can be expressed as aibj for i = 0, 1, 2, 3 and j = 0, 1. Then |G| ≤ 8 = |Q8|. Hence ϕ gives an from G to Q8.

Ex. The group defined by the generators a, b and the relations an = e, b2 = e and abab = e is the Dn.

Ex. The free group F on a set X is the group defined by the generators x ∈ X and no relations.

1We assume that elements of X can be repeated. e.g. x, y ∈ X, a relation xy−1 = e implies that x = y. 2N(Y ) is the intersection of all normal of F (X) that contain Y , or the intersection of all subgroups [ −1 of F (X) that contain xY x . x∈F (X) 1.9. FREE GROUPS, GENERATORS AND RELATIONS, AND FREE PRODUCTS 31

1.9.3 Free Products The construction of coproducts (free products) in the category of groups is similar to that of free groups. [ Let {Gi | i ∈ I} be a family of groups which are mutually disjoint sets. Let X = Gi i∈I and {1} a singleton set disjoint from X.A word on X is a sequence (a1, a2, ··· ) such that + ai ∈ X ∪ {1} and for some n ∈ Z , ai = 1 for all i ≥ n. A word (a1, a2, ··· ) is reduced provided:

1. no ai ∈ X is the identity in its group Gj;

+ 2. for any i ∈ Z , ai and ai+1 are not in the same group Gj;

3. ak = 1 implies ai = 1 for all i ≥ k. In particular 1 = (1, 1, ...) is reduced. Every reduced word (6= 1) may be written uniquely as a1a2 ··· an = (a1, a2, ··· , an, 1, 1, ··· ), where ai ∈ X. Y ∗ The set Gi of all reduced words on X forms a group, called the free group of {Gi | i∈I i ∈ I}, under the juxtaposition and necessary cancellations and contractions. For k ∈ I, there Q ∗ is a monomorphism ιk : Gk → i∈I Gi given by ek 7→ 1 and a 7→ (a, 1, 1, ··· ).

Y ∗ Thm 1.48 ( Gi is a in the category of groups). Let {Gi | i ∈ I} be a family i∈I Y ∗ of groups and Gi their . If {ψi : Gi → H | i ∈ I} is a family of group i∈I Q ∗ homomorphisms, then there exists a unique homomorphism ψ : i∈I Gi → H such that Y ∗ ψ ◦ ιi = ψi for all i ∈ I and this property determines Gi uniquely isomorphism. i∈I

Q ∗ i∈I Gi O ψ ιi # Gi / H ψi