homomorph-exercises-automorphisms.ipynb

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Section11.4Additional 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 of $G\text{,}$

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

is defined by the map

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

for $g \in G\text{.}$ 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

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

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

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

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 $\psi : k \mapsto \phi_k\text{.}$