Chapter 4.2, Problem 6E

In Exercises 1- 9, let $G$ be the given group. Write out the elements of a group of permutations that is isomorphic to $G$, and exhibit an isomorphism from $G$ to this group.

Let $G$ be the group of permutations matrices $\left\{I_3,P_1,P_2,P_3,P_4,P_5\right\}$ as given in Exercise 35 of section 3.1.

Sec $3.1,\#35\gg$

A permutation matrix is a matrix that can be obtained from an identity matrix $I_n$ by interchanging the rows one or more times (that is, by permuting the rows). For $n=3$ the permutation matrices are $I_3$ and the five matrices.

$P_1=\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right]$

$P_2=\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right]$

$P_3=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1& 0& 0\end{array}\right]$

$P_4=\left[\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right]$

$P_5=\left[\begin{array}{ccc}0& 0& 1\\ 1& 0& 0\\ 0& 1& 0\end{array}\right]$

Given that $G=\left\{I_3,P_1,P_2,P_3,P_4,P_5\right\}$ is a group of order $6$ with respect to matrix multiplication, write out a multiplication table for $G$.