alg-07-34 - canonical homomorphism Then there is a...

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Chapter 4 7. Advanced Group Theory It is important to build up the correct visions about things in a group, a homomorphism, or so. 4.1 VII-34. Isomorphism Theory Thm 4.1 (First Isomorphism Theorem). Let φ : G G be a group homomorphism with kernel K . Let γ K : G G/K be the canonical ho- momorphism defined by γ K ( g ) := gK . Then φ is the composition of two homomorphisms: φ : G γ K -→ G/K μ -→ φ [ G ] , that is, φ = μ γ K , where μ : G/K φ [ G ] is the unique isomorphism defined by μ ( gK ) := φ ( g ) . (cf. Figure 34.1 on p.307) Ex 4.2. HW 1, p.310. Below lemmas are helpful in understanding the upcoming theorems. Lem 4.3. Let N be a normal subgroup of G , and H be a subgroup of G . Then HN = NH is the smallest subgroup that contains both H and N . Moreover, if H is also normal, then HN is normal in G . Lem 4.4. Let N be a normal subgroup of G . Let γ : G G/N be the
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Unformatted text preview: canonical homomorphism. Then there is a bijection φ : { all normal subgroups of G containing N }-→ { all normal subgroups of G/N } given by φ ( L ) := γ ( L ) = LN . (By figure) 49 50 CHAPTER 4. VII Thm 4.5 (Second Isomorphism Theorem). Let N be a normal subgroup of G , and H be a subgroup of G . Then ( HN ) /N ' H/ ( H ∩ N ) . Ex 4.6 (Ex 34.6). Let G = Z × Z × Z , H = Z × Z × { } , what is ( HN ) /N ' H/ ( H ∩ N ) means? Ex 4.7. HW 3, p.310. Thm 4.8 (Third Isomorphism Theorem). Let H and K be normal subgroups of a group G with K ≤ H . Then G/H ' ( G/K ) / ( H/K ) . Ex 4.9. Hw 5, p.310. 4.1.1 Homework, VII-34, p.310-p.311 1. 1st HW: 2 , 7. 2. 2nd HW: 4, 6 , 8....
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