34 Chapter 1. Groups I (v) If G = S3 and H = (1 2) , there are exactly three left cosets of H, namely H = {(1), (1 2)} = (1 2)H, (1 3)H = {(1 3), (1 2 3)} = (1 2 3)H, (2 3)H = {(2 3), (1 3 2)} = (1 3 2)H, each of which has size two. Note that these cosets are also “parallel” that is, distinct cosets are disjoint. Consider the right cosets of H = (1 2) in S3: H = {(1), (1 2)} = H(1 2), H(1 3) = {(1 3), (1 3 2)} = H(1 3 2), H(2 3) = {(2 3), (1 2 3)} = H(1 2 3). Again, we see that there are exactly 3 (right) cosets, each of which has size two. Note that these cosets are “parallel” that is, distinct (right) cosets are disjoint. Finally, observe that the left coset (1 3)H is not a right coset of H in particular, (1 3)H = H(1 3). Lemma 1.46. Let H be a subgroup of a group G, and let a, b ∈ G. (i) aH = bH if and only if b−1a ∈ H. In particular, aH = H if and only if a ∈ H. (ii) If aH ∩ bH = ∅, then aH = bH. (iii) |aH| = |H| for all a ∈ G. Remark. Exercise 1.38 on page 37 has the version of (i) for right cosets: Ha = Hb if and only if ab−1 ∈ H, and hence Ha = H if and only if a ∈ H. Proof. The first statement follows from observing that the relation on G, defined by a ≡ b if b−1a ∈ H, is an equivalence relation21 whose equivalence classes are the left cosets. Since the equivalence classes of an equivalence relation form a partition, the left cosets of H partition G (which is the second statement). The third statement is true because h → ah is a bijection H → aH [its inverse is ah → a−1(ah)]. • For example, if H = m ⊆ Z, then a + H = b + H if and only if a − b ∈ m that is, a ≡ b mod m. The next theorem is named after Lagrange who asserted, in 1770, that the orders of certain subgroups of Sn are divisors of n!. The notion of group was invented by Galois 60 years later, and it was probably Galois who first proved the theorem in full. 21An equivalence relation on a set X is a binary relation ≡ which is reflexive, symmetric, and transitive. If a ∈ X, then its equivalence class is [a] = {x ∈ X : x ≡ a}. If ≡ is an equivalence relation on X, then the family of all equivalence classes forms a partition of X that is, they are pairwise disjoint nonempty subsets of X whose union is all of X. Conversely, given a partition (Ai)i∈I of X, there exists an equivalence relation on X whose equivalence classes are the Ai (FCAA, p. 102).

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