# Multiplicative notation if g is written

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Multiplicative notation: if G is written multiplicatively, then we write H 1 · H 2 := { h 1 h 2 : h 1 H 1 , h 2 H 2 } . Theorem 4.9 If H 1 and H 2 are subgroups of an abelian group G , then so is H 1 + H 2 . Moreover, any subgroup H of G that contains H 1 H 2 contains H 1 + H 2 , and H 1 H 2 if and only if H 1 + H 2 = H 2 . Proof. Exercise. 2 Theorem 4.10 If H 1 and H 2 are subgroups of an abelian group G , then so is H 1 H 2 . Proof. Exercise. 2 Theorem 4.11 If H 0 is a subgroup of an abelian group G , then a set H H 0 is a subgroup of G if and only if H is a subgroup of H 0 . Proof. Exercise. 2 4.3 Cosets and Quotient Groups We now generalize the notion of a congruence relation. Let G be an abelian group, and let H be a subgroup. For a, b G , we write a b (mod H ) if a - b H . It is easy to verify that the relation · ≡ · (mod H ) is an equivalence relation; that is, for all a, b, c G , we have a a (mod H ), a b (mod H ) implies b a (mod H ), and a b (mod H ) and b c (mod H ) implies a c (mod H ). Therefore, this relation partitions G into equivalence classes. It is easy to see that for any a G , the equivalence class containing a is precisely a + H := { a + h : h H } ; indeed, a b (mod H ) ⇐⇒ b - a = h for some h H ⇐⇒ b = a + h for some h H ⇐⇒ b a + H . The equivalence class a + H is called the coset of H in G containing a , and an element of such a coset is called a representative of the coset. Multiplicative notation: if G is written multiplicatively, then a b (mod H ) means a/b H , and the coset of H in G containing a is aH := { ah : h H } . 24

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Example 4.22 Let G = Z and H = n Z for some positive integer n . Then a b (mod H ) if and only if a b (mod n ). 2 Example 4.23 Let G = Z 4 and let H be the subgroup 2 Z 4 = { [0] , [2] } . The coset of H containing [1] is { [1] , [3] } . These are all the cosets of H in G . 2 Theorem 4.12 Any two cosets of a subgroup H in an abelian group G have equal cardinality; i.e., there is a bijective map from one coset to the other. Proof. Let a + H and b + H be two cosets, and consider the map f : G G that sends x G to x - a + b G . The reader may verify that f is injective and carries a + H onto b + H . 2 An incredibly useful consequence of the above theorem is: Theorem 4.13 If G is a finite abelian group, and H is a subgroup of G , then the order of H divides the order of G . Proof. This is an immediate consequence of the previous theorem, and the fact that the cosets of H in G partition G . 2 Analogous to Theorem 2.1, we have: Theorem 4.14 Let G be an abelian group and H a subgroup. For a, a 0 , b, b 0 G , if a a 0 (mod H ) and b b 0 (mod H ) , then a + b a 0 + b 0 (mod H ) . Proof. Now, a a 0 (mod H ) and b b 0 (mod H ) means that a 0 = a + h 1 and b 0 = b + h 2 for h 1 , h 2 H . Therefore, a 0 + b 0 = ( a + h 1 ) + ( b + h 2 ) = ( a + b ) + ( h 1 + h 2 ), and since h 1 + h 2 H , this means that a + b a 0 + b 0 (mod H ). 2 Let G be an abelian group and H a subgroup. Theorem 4.14 allows us to define a group operation on the collection of cosets of H in G in the following natural way: for a, b G , define ( a + H ) + ( b + H ) := ( a + b ) H.
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• Spring '13
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