PROBLEM 4 Let G be a group and let H be an abelian group. Suppose in addition that × : G → H is a group homomorphism. Define Þ : G × H → H, (g,h) = (g)h-1 VgЄ G and hЄ H. 4.1 Show that Ô is a group homomorphism. 4.2 Find the image and kernel of . 4.3 Conclude that K = {(9,4(g)) : g Є G} ≤ G × H and that (G×H)/K ≈ H.

PROBLEM 4 Let G be a group and let H be an abelian group. Suppose in addition that × : G → H is a group homomorphism. Define Þ : G × H → H, (g,h) = (g)h-1 VgЄ G and hЄ H. 4.1 Show that Ô is a group homomorphism. 4.2 Find the image and kernel of . 4.3 Conclude that K = {(9,4(g)) : g Є G} ≤ G × H and that (G×H)/K ≈ H.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.7: Direct Sums (optional)

Problem 9E: 9. Suppose that and are subgroups of the abelian group such that . Prove that .

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