Section 1.4. Lagrange’s Theorem 35 Theorem 1.47 (Lagrange’s Theorem). If H is a subgroup of a finite group G, then |H| is a divisor of |G|. Proof. Let {a1H,...,atH} be the family of all the distinct left cosets of H in G. We claim that G = a1H ∪ a2H ∪ · · · ∪ atH. If g ∈ G, then g = g1 ∈ gH but gH = aiH for some i, because a1H,...,atH is a list of all the left cosets of H. Now Lemma 1.46(ii) shows that the cosets partition G into pairwise disjoint subsets, and so |G| = |a1H| + |a2H| + · · · + |atH|. But |aiH| = |H| for all i, by Lemma 1.46(iii) hence, |G| = t|H|, as desired. • Definition. The index of a subgroup H in G, denoted by [G : H], is the number of left22 cosets of H in G. The index [G : H] is the number t in the formula |G| = t|H| in the proof of Lagrange’s Theorem, so that |G| = [G : H]|H| this formula shows that the index [G : H] is also a divisor of |G| moreover, [G : H] = |G|/|H|. Example 1.48. (i) Here is another solution of Exercise 1.22 on page 16. In Example 1.45(iv), we saw that Sn = An ∪ τAn, where τ is a transposition. Thus, there are exactly two cosets of An in Sn that is, [Sn : An] = 2. It follows that |An| = 1 2 n!. (ii) Recall that the dihedral group D2n = Σ(πn), the symmetries of the regular n-gon πn, has order 2n, and it contains the cyclic subgroup ρ of order n generated by the clockwise rotation ρ by (360/n)o. Thus, ρ has index [D2n : ρ ] = 2n/n = 2, and there are only two cosets: ρ and σ ρ , where σ is any reflection outside of ρ . It follows that D2n = ρ ∪ σ ρ every element α ∈ D2n has a unique factorization α = σiρj, where i = 0, 1 and 0 ≤ j n. Corollary 1.49. If G is a finite group and a ∈ G, then the order of a is a divisor of |G|. Proof. Immediate from Lagrange’s Theorem, for the order of a is | a |. • Corollary 1.50. If G is a finite group, then a|G| = 1 for all a ∈ G. Proof. If a has order d, then |G| = dm for some integer m, by the previous corollary, and so a|G| = adm = (ad)m = 1. • Corollary 1.51. If p is prime, then every group G of order p is cyclic. 22Exercise 1.44 on page 38 shows that the number of left cosets of a subgroup H is equal to the number of right cosets of H.

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