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Let $G$ be a group. Let $a ∈ G$. An inner automorphism of $G$ is a

function of the form $\gamma_a : G → G$ given by $\gamma_a(g) = aga^{-1}$.

Let $Inn(G)$ be the set of all inner automorphisms of G.

(a) Prove that $Inn(G)$ forms a group. (starting by identifying an appropriate binary operation.)

(b) Deﬁne $\varphi : G → Inn(G)$ by $\varphi(a) = \varphi_a$. Verify that $\varphi$ is surjective homomorphism and identify the kernel of $\varphi$.

function of the form $\gamma_a : G → G$ given by $\gamma_a(g) = aga^{-1}$.

Let $Inn(G)$ be the set of all inner automorphisms of G.

(a) Prove that $Inn(G)$ forms a group. (starting by identifying an appropriate binary operation.)

(b) Deﬁne $\varphi : G → Inn(G)$ by $\varphi(a) = \varphi_a$. Verify that $\varphi$ is surjective homomorphism and identify the kernel of $\varphi$.

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