##### 1

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, $\aut(G) \leq S_G$.

##### 2

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 $i_g \in \aut(G)$.

##### 3

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

##### 4

Find an automorphism of a group $G$ that is not an inner automorphism.

##### 5

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*}

##### 6

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

##### 7

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

##### 8

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

##### 9

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.

##### 10

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

##### 11

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

##### 12

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