Paper 3, Section II, D

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

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

Prove that the image of $\theta$ is a normal subgroup of $A$.

Show that if $G$ is cyclic then $A$ is abelian. If $G$ is abelian, must $A$ be abelian? Justify your answer.

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