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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, VII34, p.310p.311 1. 1st HW: 2 , 7. 2. 2nd HW: 4, 6 , 8....
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 Spring '08
 Staff
 Algebra, Group Theory, Normal subgroup, Homomorphism, Group homomorphism, Third Isomorphism Theorem

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