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Let \(\aut(G)\) be the set of all automorphisms of \(G\); that is, isomorphisms from \(G\) to itself. Prove this set forms a group and is a subgroup of the group of permutations of \(G\); that is, \(\aut(G) \leq S_G\).

Let \(\aut(G)\) be the set of all automorphisms of \(G\); that is, isomorphisms from \(G\) to itself. Prove this set forms a group and is a subgroup of the group of permutations of \(G\); that is, \(\aut(G) \leq S_G\).

An *inner automorphism* of \(G\),
\begin{equation*}i_g : G \rightarrow G,\end{equation*}
is defined by the map
\begin{equation*}i_g(x) = g x g^{-1},\end{equation*}
for \(g \in G\). Show that \(i_g \in \aut(G)\).

The set of all inner automorphisms is denoted by \(\inn(G)\). Show that \(\inn(G)\) is a subgroup of \(\aut(G)\).

Find an automorphism of a group \(G\) that is not an inner automorphism.

Let \(G\) be a group and \(i_g\) be an inner automorphism of \(G\), and define a map \begin{equation*}G \rightarrow \aut(G)\end{equation*} by \begin{equation*}g \mapsto i_g.\end{equation*} Prove that this map is a homomorphism with image \(\inn(G)\) and kernel \(Z(G)\). Use this result to conclude that \begin{equation*}G/Z(G) \cong \inn(G).\end{equation*}

Compute \(\aut(S_3)\) and \(\inn(S_3)\). Do the same thing for \(D_4\).

Find all of the homomorphisms \(\phi : {\mathbb Z} \rightarrow {\mathbb Z}\). What is \(\aut({\mathbb Z})\)?

Find all of the automorphisms of \({\mathbb Z}_8\). Prove that \(\aut({\mathbb Z}_8) \cong U(8)\).

For \(k \in {\mathbb Z}_n\), define a map \(\phi_k : {\mathbb Z}_n \rightarrow {\mathbb Z}_n\) by \(a \mapsto ka\). Prove that \(\phi_k\) is a homomorphism.

Prove that \(\phi_k\) is an isomorphism if and only if \(k\) is a generator of \({\mathbb Z}_n\).

Show that every automorphism of \({\mathbb Z}_n\) is of the form \(\phi_k\), where \(k\) is a generator of \({\mathbb Z}_n\).

Prove that \(\psi : U(n) \rightarrow \aut({\mathbb Z}_n)\) is an isomorphism, where \(\psi : k \mapsto \phi_k\).