Section 1.6. Quotient Groups 49 We now show that HK is a subgroup. Since 1 H and 1 K, we have 1 = 1 · 1 HK if hk HK, then (hk)−1 = k−1h−1 KH = HK if hk, h1k1 HK, then hkh1k1 HKHK = HHKK = HK. (ii) If g G, then Lemma 1.71 gives gHK = HgK = HKg, and the same lemma now gives HK G. Here is a fundamental construction of a new group from a given group. Theorem 1.73. Let G/K denote the family of all the left cosets of a subgroup K of G. If K is a normal subgroup, then aKbK = abK for all a, b G, and G/K is a group under this operation. Proof. Generalized associativity holds in S(G), by Corollary 1.25, because it is a semigroup. Thus, we may view the product of two cosets (aK)(bK) as the product {a}K{b}K of four elements in S(G): (aK)(bK) = a(Kb)K = a(bK)K = abKK = abK normality of K gives Kb = bK for all b K (Lemma 1.71), while KK = K (because K is a subgroup). Hence, the product of two cosets of K is again a coset of K, and so a binary operation on G/K has been defined. As multiplication in S(G) is associative, so, in particular, is the multiplication of cosets in G/K. The identity is the coset K = 1K, for (1K)(bK) = 1bK = bK = b1K = (bK)(1K), and the inverse of aK is a−1K, for (a−1K)(aK) = a−1aK = K = aa−1K = (aK)(a−1K). Therefore, G/K is a group. It is important to remember what we have just proved: the product aKbK = abK in G/K does not depend on the particular representatives of the cosets. Thus, the law of substitution holds: if aK = a K and bK = b K, then abK = aKbK = a Kb K = a b K. Definition. The group G/K is called the quotient group G mod K. When G is finite, its order |G/K| is the index [G : K] = |G|/|K| (presumably, this is the reason why quotient groups are so called). Example 1.74. We show that the quotient group G/K is precisely Im when G is the additive group Z and K = m , the (cyclic) subgroup of all the multiples of a positive integer m. Since Z is abelian, m is necessarily a normal subgroup. The sets Z/ m and Im coincide because they are comprised of the same elements the coset a + m is the congruence class [a]: a + m = {a + km : k Z} = [a]. The binary operations also coincide: addition in Z/ m is given by (a + m ) + (b + m ) = (a + b) + m since a + m = [a], this last equation is just [a] + [b] = [a + b], which is the sum in Im. Therefore, Im and the quotient group Z/ m are equal (and not merely isomorphic).
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