Math 360, Fall 2021, Assignment 10

I have found a very great number of exceedingly beautiful theorems."

- Pierre de Fermat

Carefully define the following terms, then give one example and one non-example of each:

1. Permutation model (for a group $G$).
2. Finite permutation model (for a group $G$).

Carefully state the following theorems (you do not need to prove them):

1. Cayley's Theorem.

Solve the following problems:

1. Our proof of Cayley's Theorem was constructive, meaning that we proved that every group has a permutation model by actually constructing and exhibiting one specific permutation model. The particular permutation model constructed in our proof is called the Cayley model, or in some contexts, the left regular representation. Explicitly write down the Cayley model of the dihedral group $D_4$. (That is, explicitly write the permutation assigned by the Cayley model to each of the eight elements of $D_4$.)
2. Now show that permutation models are not unique, by writing down a second and different permutation model for $D_4$. (Hint: begin by numbering the vertices of a square.)

--------------------End of assignment--------------------

Definitions:

1. Permutation model (for a group $G$): Let $G$ be any group. A permutation model for $G$ is an isomorphism from $G$ to some subgroup of $Sym(S)$.
2. Finite permutation model (for a group $G$): For some set $S$. If S is finite, this is said to be a finite permutation model.

Theorems:

1. Cayley's Theorem: Every Group $G$ has at least one permutation model. If $G$ is finite, it has at least one finite permutation model.

Other Problems:

1. $\begin{pmatrix}1&2&3&4\\1&2&3&4\end{pmatrix}$, $\begin{pmatrix}1&2&3&4\\2&3&1&4\end{pmatrix}$, $\begin{pmatrix}1&2&3&4\\3&4&1&2\end{pmatrix}$, $\begin{pmatrix}1&2&3&4\\4&1&2&3\end{pmatrix}$, $\begin{pmatrix}1&2&3&4\\3&2&1&4\end{pmatrix}$, $\begin{pmatrix}1&2&3&4\\1&4&3&2\end{pmatrix}$, $\begin{pmatrix}1&2&3&4\\4&3&2&1\end{pmatrix}$, $\begin{pmatrix}1&2&3&4\\2&1&4&3\end{pmatrix}$
2. I DONT KNOW