Abstract Algebra: Theory and Applications

Let $\mathrm{Aut}(G)$ 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, $\mathrm{Aut}(G)\le S_G\text{.}$

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

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

Compute $\mathrm{Aut}(S_3)$ and $\mathrm{Inn}(S_3)\text{.}$ Do the same thing for $D_4\text{.}$

Find all of the homomorphisms $\varphi :\mathbb{Z}\to \mathbb{Z}\text{.}$ What is $\mathrm{Aut}(\mathbb{Z})\text{?}$

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

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

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

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

Prove that $\psi :U(n)\to \mathrm{Aut}({\mathbb{Z}}_n)$ is an isomorphism, where $\psi :k\mapsto {\varphi }_k\text{.}$