Let G1 and G2 be groups and let ƒ: G₁ → G2 be a homomorphism. (i) Show that, if H2 is a normal subgroup of G2, then ƒ¯¹ (H2) is a normal subgroup of G1. (ii) Show that, if H₁ is a normal subgroup of G₁, then f(H₁) need not be a normal subgroup of G2. (iii) Show that, if H₁ is a normal subgroup of G₁, and ƒ is surjective, then ƒ(H₁) is a normal subgroup of G2.

Let G1 and G2 be groups and let ƒ: G₁ → G2 be a homomorphism. (i) Show that, if H2 is a normal subgroup of G2, then ƒ¯¹ (H2) is a normal subgroup of G1. (ii) Show that, if H₁ is a normal subgroup of G₁, then f(H₁) need not be a normal subgroup of G2. (iii) Show that, if H₁ is a normal subgroup of G₁, and ƒ is surjective, then ƒ(H₁) is a normal subgroup of G2.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.6: Quotient Groups

Problem 8E: Suppose G1 and G2 are groups with normal subgroups H1 and H2, respectively, and with G1/H1...

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