18 Chapter 1. Groups I Definition. A group G is called abelian10 if it satisfies the commutative law: x ∗ y = y ∗ x for every x, y ∈ G. The groups Sn, for n ≥ 3, are not abelian because (1 2) and (1 3) are elements of Sn that do not commute: (1 2)(1 3) = (1 3 2) and (1 3)(1 2) = (1 2 3). Lemma 1.16. Let G be a group. (i) The cancellation laws hold: if either x ∗ a = x ∗ b or a ∗ x = b ∗ x, then a = b.11 (ii) The element e is the unique element in G with e∗x = x = x∗e for all x ∈ G. (iii) Each x ∈ G has a unique inverse: there is only one element x ∈ G with x ∗ x = e = x ∗ x (henceforth, this element will be denoted by x−1). (iv) (x−1)−1 = x for all x ∈ G. Proof. (i) Choose x with x ∗ x = e = x ∗ x then a = e ∗ a = (x ∗ x) ∗ a = x ∗ (x ∗ a) = x ∗ (x ∗ b) = (x ∗ x) ∗ b = e ∗ b = b. A similar proof works when x is on the right. (ii) Let e0 ∈ G satisfy e0 ∗ x = x = x ∗ e0 for all x ∈ G. In particular, setting x = e in the second equation gives e = e ∗ e0 on the other hand, the defining property of e gives e ∗ e0 = e0, so that e = e0. (iii) Assume that x ∈ G satisfies x ∗ x = e = x ∗ x. Multiply the equation e = x ∗ x on the left by x to obtain x = x ∗ e = x ∗ (x ∗ x ) = (x ∗ x) ∗ x = e ∗ x = x . (iv) By definition, (x−1)−1 ∗ x−1 = e = x−1 ∗ (x−1)−1. But x ∗ x−1 = e = x−1 ∗ x, so that (x−1)−1 = x, by (iii). • From now on, we will usually denote the product x ∗ y in a group by xy, and we will denote the identity by 1 instead of by e. When a group is abelian, however, we usually use the additive notation x + y in this case, the identity is denoted by 0, and the inverse of an element x is denoted by −x instead of by x−1. Example 1.17. (i) The set Q× of all nonzero rationals is an abelian group, where ∗ is ordinary multiplication: the number 1 is the identity, and the inverse of r ∈ Q× is 1/r. Similarly, R× and C× are multiplicative abelian groups. 10Commutative groups are called abelian because Abel proved (in modern language) that if the Galois group of a polynomial f(x) is commutative, then f(x) is solvable by radicals. 11We cannot cancel x if x ∗ a = b ∗ x. For example, we have (1 2)(1 2 3) = (2 1 3)(1 2) in S3, but (1 2 3) = (2 1 3).

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