## Exercises11.5Additional Exercises: Automorphisms

### 1.

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

### 2.

An inner automorphism of $$G\text{,}$$

\begin{equation*} i_g : G \rightarrow G\text{,} \end{equation*}

is defined by the map

\begin{equation*} i_g(x) = g x g^{-1}\text{,} \end{equation*}

for $$g \in G\text{.}$$ Show that $$i_g \in \aut(G)\text{.}$$

### 3.

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

### 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 $$G\text{,}$$ and define a map

\begin{equation*} G \rightarrow \aut(G) \end{equation*}

by

\begin{equation*} g \mapsto i_g\text{.} \end{equation*}

Prove that this map is a homomorphism with image $$\inn(G)$$ and kernel $$Z(G)\text{.}$$ Use this result to conclude that

\begin{equation*} G/Z(G) \cong \inn(G)\text{.} \end{equation*}

### 6.

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

### 7.

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

### 8.

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

### 9.

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

### 10.

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

### 11.

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

### 12.

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