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).

Q: ii) Does there exist a group G such that G/[G,G] is non-abelian? Give an example, or prove

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Q: (3) For any group G, Z(G) ≤ [G, G].

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Q: 6. Let G be a group. An isomorphism from G to G is called an automorphism of G. The set of all…

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