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).
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).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![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 a
group under composition.
(ii) Prove that y : G → Aut(G), defi ned by g → ½ (conjugation by g), is a
homomorphism.
(iii) Prove that ker y = Z(G).
(iv) Prove that imy 4 Aut(G).
35](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6554676b-d841-43d3-9007-f2a3484be24a%2F8a098cb8-812b-4a5b-bb5b-4cb15fec1e15%2F5c5fuzo_processed.jpeg&w=3840&q=75)
Transcribed Image Text: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 a
group under composition.
(ii) Prove that y : G → Aut(G), defi ned by g → ½ (conjugation by g), is a
homomorphism.
(iii) Prove that ker y = Z(G).
(iv) Prove that imy 4 Aut(G).
35
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