PROBLEM 3 3.1 Let G be a group, and recall that Sym(G) is the group of invertible functions G → G. Use the Subgroup Criterion to show that Aut (G) ≤ Sym(G), where Aut (G){Isomorphisms : G→ G}. 3.2 Let Є Aut(Z). Prove by induction on k that (k) = kó(1) for all k € Z. Afterwards, use properties of group homomorphisms to conclude that this identity in fact holds for all k Є Z. 3.3 Let Q denote the multiplicative group of nonzero rational numbers. Show § : Aut(G) → Q*, §(x) = x(1) VЄ Aut(G) is an injective group homomorphism, and find Im(Þ). Hint: For Im(), use the identity from 3.2 with k - 6¯¹(1) to see that ¯¹(1)(1) = 1. This means (1) is an integer with an integer inverse. What does this mean about Im()?

PROBLEM 3 3.1 Let G be a group, and recall that Sym(G) is the group of invertible functions G → G. Use the Subgroup Criterion to show that Aut (G) ≤ Sym(G), where Aut (G){Isomorphisms : G→ G}. 3.2 Let Є Aut(Z). Prove by induction on k that (k) = kó(1) for all k € Z. Afterwards, use properties of group homomorphisms to conclude that this identity in fact holds for all k Є Z. 3.3 Let Q denote the multiplicative group of nonzero rational numbers. Show § : Aut(G) → Q*, §(x) = x(1) VЄ Aut(G) is an injective group homomorphism, and find Im(Þ). Hint: For Im(), use the identity from 3.2 with k - 6¯¹(1) to see that ¯¹(1)(1) = 1. This means (1) is an integer with an integer inverse. What does this mean about Im()?

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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