Paper 3, Section II, D

Groups | Part IA, 2018

Let gg be an element of a group GG. We define a map gg^{*} from GG to GG by sending xx to gxg1g x g^{-1}. Show that gg^{*} is an automorphism of GG (that is, an isomorphism from GG to GG ).

Now let AA denote the group of automorphisms of GG (with the group operation being composition), and define a map θ\theta from GG to AA by setting θ(g)=g\theta(g)=g^{*}. Show that θ\theta is a homomorphism. What is the kernel of θ\theta ?

Prove that the image of θ\theta is a normal subgroup of AA.

Show that if GG is cyclic then AA is abelian. If GG is abelian, must AA be abelian? Justify your answer.

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