An automorphism of a group G is an isomorphism G → G.(i) Prove that Aut(G), the set of all the automorphisms of a group G, is agroup under composition.(ii) Prove that y : G → Aut(G), defi ned by g → ½ (conjugation by g), is ahomomorphism.(iii) Prove that ker y = Z(G).(iv) Prove that imy 4 Aut(G).35

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Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.5: Isomorphisms

Problem 1E: Prove that if is an isomorphism from the group G to the group G, then 1 is an isomorphism from G to...

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An automorphism of a group G is an isomorphism G→ G. (1) Prove that Aut(G), the set of all the automorphisms of a group G, is a group under composition. (ii) Prove that y: G→ Aut(G), defined by g (conjugation by g), is a homomorphism. (iii) Prove that ker y = Z(G).