32 Chapter 1. Groups I If X is a nonempty subset of a group G, a word 20 on X is an element g ∈ G of the form g = x11 e · · · xnn e , where xi ∈ X and ei = ±1 for all i. Proposition 1.42. If X is a nonempty subset of a group G, then X is the set of all the words on X. Proof. We claim that W (X), the set of all the words on X, is a subgroup. If x ∈ X, then 1 = xx−1 ∈ W (X) the product of two words on X is also a word on X the inverse of a word on X is a word on X. It now follows that X ⊆ W (X), for W (X) is a subgroup containing X. The reverse inclusion is clear, for any subgroup of G containing X must contain every word on X. Therefore, X = W (X). • Definition. If H and K are subgroups of a group G, then H ∨ K = H ∪ K is the subgroup generated by H and K. It is easy to check that H ∨ K is the smallest subgroup of G that contains both H and K. Corollary 1.43. If H and K are subgroups of an abelian group G, then H ∨ K = H + K = {sh + tk : h ∈ H, k ∈ K, s, t ∈ Z}. Proof. Words x11 e · · · xnn e ∈ H ∪ K are written e1x1 + · · · + enxn in additive notation, and they can be written in the displayed form because G’s being abelian allows us to collect terms. • Example 1.44. (i) If G = a is a cyclic group with generator a, then G is generated by the subset X = {a}. (ii) Let a and b be integers, and let A = a and B = b be the cyclic subgroups of Z they generate. Then A ∩ B = m , where m = lcm{a, b}, and A + B = d , where d is the gcd (a, b). (iii) The dihedral group D2n (the symmetry group of a regular n-gon, where n ≥ 3) is generated by ρ, σ, where ρ is a rotation by (360/n)o and σ is a reflection. Note that these generators satisfy the equations ρn = 1, σ2 = 1, and σρσ = ρ−1. We defined the dihedral group D4 = V, the four-group, in Example 1.33(i) note that V is generated by elements ρ and σ satisfying the equations ρ2 = 1, σ2 = 1, and σρσ = ρ−1 = ρ. Perhaps the most fundamental fact about subgroups H of a finite group G is that their orders are constrained. Certainly, we have |H| ≤ |G|, but it turns out that |H| must be a divisor of |G|. To prove this, we introduce the notion of coset. Definition. If H is a subgroup of a group G and a ∈ G, then the coset aH is the subset aH of G, where aH = {ah : h ∈ H}. 20This term will be modified a bit when we discuss free groups in Chapter 4.

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