Question

the following questions are relative,please solve them, thanks!

4. Let G be a group. An isomorphism : G G is called an automorphism of G. (a) Prove that the set, Aut(G), of all automorphisms of G forms a group under composition. (b) Let g E G. Show that the map ф9: G-+ G given by c%(z)-gZg", įs an automorphism. These are called the inner automorphisms of G (c) Show that the set of all g E G such that Og-Pe is a normal subgroup of G. Show that H is a normal subgroup of G if and only if φ,(H)-H. for all inner automorphisms φ, of G (as defined in Question (4b) above). Let G be a group. Show that two elements a,y E G (respectively, two subgroups H, K in G) are conjugate if and only if there exists an inner automorphism 0, as defined in Question (4b) above, such that ф9(z) y (respectively, ф9(H)-K)

Answer 1

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