Question

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):

(b)$\mathrm{\mathbb{Z}}$₄

Short answer

Answer

The left representation of is where,$$\begin{gathered} {\mathrm{\mathbb{Z}}}_{4}isA=\{f_{0,}{f}_1,f_2,f_3\}where,f_0=\left(\begin{array}{cccc}0& 1& 2& 3\\ 0& 1& 2& 3\end{array}\right),f_1=\left(\begin{array}{cccc}0& 1& 2& 3\\ 1& 2& 2& 0\end{array}\right), \\ {f}_2=\left(\begin{array}{cccc}0& 1& 2& 3\\ 2& 3& 0& 1\end{array}\right),and{f}_3=\left(\begin{array}{cccc}0& 1& 2& 3\\ 3& 0& 1& 2\end{array}\right), \end{gathered}$$

Step-by-step solution

Left representation of Group

$$\begin{gathered} SupposethatGisagroup,A=f_g:g\in Gthegroupwhere{f}_g:g\to Gisdefinedby \\ {f}_g\left(x\right)=g\left(x\right)orallx\in G.GroupiscalledtheleftrepresentationofG. \end{gathered}$$

Find the left regular representation of group  

Consider the group ${\mathrm{\mathbb{Z}}}_4=\{0,1,2,3\}$.

Define by

,$$\begin{gathered} f_0={\mathrm{\mathbb{Z}}}_4\to {\mathrm{\mathbb{Z}}}_{4}by \\ {f}_0\left(x\right)=0{\oplus }_{4}xforallx\in {\mathrm{\mathbb{Z}}}_4 \end{gathered}$$

Find the image of the elements of ${\mathrm{\mathbb{Z}}}_4=\{0,1,2,3\}$under the function$f_0$.

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

Therefore, the function$f_0$is represented aslocalid="1651597207376" $f_0=\left(\begin{array}{cccc}0& 1& 2& 3\\ 0& 1& 2& 3\end{array}\right)$.

Define by,

$$\begin{gathered} f_1={\mathrm{\mathbb{Z}}}_4\to {\mathrm{\mathbb{Z}}}_{4}by \\ {f}_1\left(x\right)=0{\oplus }_{4}xforallx\in {\mathrm{\mathbb{Z}}}_4 \end{gathered}$$

Find the image of the elements ${\mathrm{\mathbb{Z}}}_4=\{0,1,2,3\}$of under the function$f_1$.

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

Therefore, the function $f_1$is represented as $f_1=\left(\begin{array}{cccc}0& 1& 2& 3\\ 1& 2& 3& 0\end{array}\right)$.

Define by,

$$\begin{gathered} f_2={\mathrm{\mathbb{Z}}}_4\to {\mathrm{\mathbb{Z}}}_{4}by \\ {f}_2\left(x\right)=2{\oplus }_{4}xforallx\in {\mathrm{\mathbb{Z}}}_4 \end{gathered}$$

Find the image of the elements ${\mathrm{\mathbb{Z}}}_4=\{0,1,2,3\}$of under the function $f_2$.

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

Therefore, the function $f_2$is represented as $f_2=\left(\begin{array}{cccc}0& 1& 2& 3\\ 2& 3& 0& 1\end{array}\right)$

Define

$$\begin{gathered} f_3={\mathrm{\mathbb{Z}}}_4\to {\mathrm{\mathbb{Z}}}_{4}by \\ {f}_3\left(x\right)=3{\oplus }_{4}xforallx\in {\mathrm{\mathbb{Z}}}_4 \end{gathered}$$

Find the image of the elements ${\mathrm{\mathbb{Z}}}_4=\{0,1,2,3\}$of under the function $f_3$.

$$\begin{gathered} f_3\left(0\right)=3{\oplus }_{3}0=2 \\ {f}_3\left(1\right)=3{\oplus }_{3}1=0 \\ {f}_3\left(2\right)=3{\oplus }_{3}2=1 \\ {f}_3\left(3\right)=3{\oplus }_{3}3=2 \end{gathered}$$

Therefore, the function $f_3$is represented as .$f_3=\left(\begin{array}{cccc}0& 1& 2& 3\\ 3& 0& 1& 2\end{array}\right)$

.

Therefore the left representation

$$\begin{gathered} {\mathrm{\mathbb{Z}}}_{4}isA=\{f_{0,}{f}_1,f_2,f_3\}where,f_0=\left(\begin{array}{cccc}0& 1& 2& 3\\ 0& 1& 2& 3\end{array}\right),f_1=\left(\begin{array}{cccc}0& 1& 2& 3\\ 1& 2& 2& 0\end{array}\right), \\ {f}_2=\left(\begin{array}{cccc}0& 1& 2& 3\\ 2& 3& 0& 1\end{array}\right),and{f}_3=\left(\begin{array}{cccc}0& 1& 2& 3\\ 3& 0& 1& 2\end{array}\right), \end{gathered}$$