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):

(c) S₃

Short answer

Answer

Therepresentation of S₃is $A=\{f_{a,}{f}_{b,}{f}_{c,}{f}_d,f_e{f}_f\}$.

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 S3

Consider the group S₃= { (1), (23),(13),(12),(123),(132)}

The group $f_g:g\in S_3$is called the left representation of S₃ .

Let,

a = (1)

b = (23)

c = (13)

d = (12)

And,

e = (123)

f = (132)

The image of the elements S₃of under the function f_g are the respective elements of the row headed by g.

Therefore, the left representations are as follows:

$$\begin{gathered} f_{\left(1\right)}=f_a=\left(\begin{array}{cccccc}a& b& c& d& e& f\\ a& b& c& d& e& f\end{array}\right) \\ {f}_{\left(23\right)}=f_b=\left(\begin{array}{cccccc}a& b& c& d& e& f\\ b& a& e& f& c& d\end{array}\right) \\ {f}_{\left(13\right)}=f_c=\left(\begin{array}{cccccc}a& b& c& d& e& f\\ c& f& a& e& d& b\end{array}\right) \\ {f}_{\left(12\right)}=f_d=\left(\begin{array}{cccccc}a& b& c& d& e& f\\ d& e& f& a& b& c\end{array}\right) \\ And, \\ {f}_{\left(123\right)}=f_e=\left(\begin{array}{cccccc}a& b& c& d& e& f\\ e& d& b& c& f& a\end{array}\right) \\ {f}_{\left(132\right)}=f_f=\left(\begin{array}{cccccc}a& b& c& d& e& f\\ f& c& d& b& a& e\end{array}\right) \end{gathered}$$

Therefore, the representation of S₃ is$A=\{f_{a,}{f}_{b,}{f}_{c,}{f}_d,f_e{f}_f\}$