Textbook Problem

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.

(Sec. $3.3,\#22c,32c$, Sec. $3.4,\#5$, Sec. $4.2,\#6$)

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

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

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

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

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

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 $G$.

Sec. $3.3,\#22c,32c\ll$

22. Find the center $Z(G)$ for each of the following groups $G$.

c. $G=\left\{I_3,P_1,P_2,P_3,P_4,P_5\right\}$ in Exercise 35 of section 3.1.

32. Find the centralizer for each element $a$ in each of the following groups.

c. $G=\left\{I_3,P_1,P_2,P_3,P_4,P_5\right\}$ in Exercise 35 of section 3.1

Sec. $3.4,\#5\ll$

5. The elements of the multiplicative group $G$ of $3\times 3$ permutation matrices are given in Exercise $35$ of section $3.1$. Find the order of each element of the group.

Sec. $4.2,\#6\ll$

6. 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 $3.1$.