Section 1.4. Lagrange’s Theorem 29 Consider what verifying associativity a(bc) = (ab)c would involve: there are four choices for each of a, b, and c, and so there are 43 = 64 equations to be checked. (ii) If R2 is the plane considered as an (additive) abelian group [see Exam- ple 1.17(v)], then any line L through the origin is a subgroup. The easiest way to see this is to choose a point (a, b) = (0, 0) on L and then note that L consists of all the scalar multiples (ra, rb). The reader may now verify that the axioms in the definition of subgroup do hold for L. (iii) The group μn of nth roots of unity [see Example 1.17(iv)] is a subgroup of the circle group S1, but it is not a subgroup of the plane R2. We can shorten the list of items needed to verify that a subset is, in fact, a subgroup. Proposition 1.34. A subset H of a group G is a subgroup if and only if H is nonempty and xy−1 ∈ H whenever x, y ∈ H. Proof. Necessity is clear. For suﬃciency, take x ∈ H (which exists because H = ∅) by hypothesis, 1 = xx−1 ∈ H. If y ∈ H, then y−1 = 1y−1 ∈ H, and if x, y ∈ H, then xy = x(y−1)−1 ∈ H. • Note that if the binary operation on G is addition, then the condition in the proposition is that H is a nonempty subset such that x, y ∈ H implies x − y ∈ H. Of course, the simplest way to check that a candidate H for a subgroup is nonempty is to check whether 1 ∈ H. Corollary 1.35. A nonempty subset H of a finite group G is a subgroup if and only if H is closed that is, x, y ∈ H implies xy ∈ H. Proof. Since G is finite, Proposition 1.29 says that each x ∈ G has finite order. Hence, if xn = 1, then 1 ∈ H and x−1 = xn−1 ∈ H. • This corollary can be false when G is an infinite group. For example, let G be the additive group Z the set N = {0, 1, 2,... } of natural numbers is closed under addition, but N is not a subgroup of Z. Example 1.36. The subset An = {α ∈ Sn : α is even} ⊆ Sn is a subgroup, by Proposition 1.12, for it is closed under multiplication: even ◦ even = even. The group An is called the alternating group17 on n letters. Definition. If G is a group and a ∈ G, then the cyclic subgroup of G generated by a, denoted by a , is a = {an : n ∈ Z} = {all powers of a}. 17The alternating group first arose in studying polynomials. If f(x) = (x − u1)(x − u2) · · · (x − un), where u1,...,un are distinct, then the number D = ij (ui − uj) can change sign when the roots are permuted: if α is a permutation of {u1, u2,...,un}, then ij [α(ui) − α(uj)] = ±D. Thus, the sign of the product alternates as various permutations α are applied to its factors. The sign does not change for those α in the alternating group.

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