Question

Find the left regular representation of each group (that is, express each group as a permutation group as in the proof of Theorem 7.21):

(a)${\mathrm{\mathbb{Z}}}_3$

Short answer

The left representation of ${\mathrm{\mathbb{Z}}}_3$is $A=\left\{f_0,f_1,f_2\right\}$ where, $f_0=\left(\begin{array}{ccc}0& 1& 2\\ 0& 1& 2\end{array}\right),$$f_1=\left(\begin{array}{ccc}0& 1& 2\\ 1& 2& 0\end{array}\right),$ and $f_2=\left(\begin{array}{ccc}0& 1& 2\\ 2& 0& 1\end{array}\right)$.

Step-by-step solution

Left representation of Group

Suppose that G is a group, the group$\mathit{A}=\left\{f_g:g\in G\right\}$ where${\mathit{f}}_{\mathbf{g}}\mathbf{}\mathbf{:}\mathbf{}\mathit{G}\mathbf{\to }\mathit{G}$ is defined by${\mathit{f}}_g\left(x\right)\mathbf{=}\mathit{g}\mathit{x}$ for all $\mathit{x}\mathit{\in }\mathit{G}$. The group$\mathit{A}=\left\{f_g:g\in G\right\}$ is called the left representation of G.

Find the left regular representation of group ℤ3

Consider the group ${\mathrm{\mathbb{Z}}}_3=\left\{0,1,2\right\}$.

Define $f_0:{\mathrm{\mathbb{Z}}}_3\to {\mathrm{\mathbb{Z}}}_3$by,

$f_0\left(x\right)=0{\oplus }_{3}x$ for all$x\in {\mathrm{\mathbb{Z}}}_3$

Find the image of the elements of${\mathrm{\mathbb{Z}}}_3=\left\{0,1,2\right\}$ under the function $f_0$.

$$\begin{gathered} f_0\left(0\right)=0{\oplus }_{3}0=0 \\ {f}_0\left(1\right)=0{\oplus }_{3}1=1 \\ {f}_0\left(2\right)=0{\oplus }_{3}2=2 \end{gathered}$$

Therefore, the function$f_0$ is represented as $f_0=\left(\begin{array}{ccc}0& 1& 2\\ 0& 1& 2\end{array}\right)$.

Define$f_1:{\mathrm{\mathbb{Z}}}_3\to {\mathrm{\mathbb{Z}}}_3$ by,

$f_1\left(x\right)=1{\oplus }_{3}x$ for all$x\in {\mathrm{\mathbb{Z}}}_3$

Find the image of the elements of${\mathrm{\mathbb{Z}}}_3=\left\{0,1,2\right\}$ under the function $f_1$.

$$\begin{gathered} f_1\left(0\right)=1{\oplus }_{3}0=1 \\ {f}_1\left(1\right)=1{\oplus }_{3}1=2 \\ {f}_1\left(2\right)=1{\oplus }_{3}2=0 \end{gathered}$$

Therefore, the function$f_1$ is represented as $f_1=\left(\begin{array}{ccc}0& 1& 2\\ 1& 2& 0\end{array}\right)$.

Define$f_2:{\mathrm{\mathbb{Z}}}_3\to {\mathrm{\mathbb{Z}}}_3$ by,

$f_2\left(x\right)=2{\oplus }_{3}x$ for all$x\in {\mathrm{\mathbb{Z}}}_3$

Find the image of the elements of${\mathrm{\mathbb{Z}}}_3=\left\{0,1,2\right\}$ under the function $f_2$.

$$\begin{gathered} f_2\left(0\right)=2{\oplus }_{3}0=2 \\ {f}_2\left(1\right)=2{\oplus }_{3}1=0 \\ {f}_2\left(2\right)=2{\oplus }_{3}2=1 \end{gathered}$$

Therefore, the function$f_2$ is represented as $f_2=\left(\begin{array}{ccc}0& 1& 2\\ 2& 0& 1\end{array}\right)$.

Therefore, the left representation of ${\mathrm{\mathbb{Z}}}_3$is $A=\left\{f_0,f_1,f_2\right\}$where,$f_0=\left(\begin{array}{ccc}0& 1& 2\\ 0& 1& 2\end{array}\right),$$f_1=\left(\begin{array}{ccc}0& 1& 2\\ 1& 2& 0\end{array}\right),$ and $f_2=\left(\begin{array}{ccc}0& 1& 2\\ 2& 0& 1\end{array}\right)$.