Exercise 1.7.13 (Kernel of the left regular action)

Question

Find the kernel of the left regular action.

Richard Ganaye · Accepted Answer

\begin{proof} 
The left regular action is the map
$$
\left\{
\begin{array}{ccl}
G 	imes G & 	o & G\
(g,a) & \mapsto & g\cdot a = ga.
\end{array}
ight.
$$
The corresponding permutation representation is
$$
\varphi
\left\{
\begin{array}{ccl}
G & 	o &S_G\
g & \mapsto &\varphi_g,
\end{array}
ight.
\qquad 	ext{where} \qquad \varphi_g
\left\{
\begin{array}{ccl}
G & 	o &G\
g & \mapsto &\varphi_g(a) = ga.
\end{array}
ight.
$$
If $g \in G$, then $\varphi_g = \mathrm{Id}_G$ if and only if for all $a \in G$, $ga = a$. In particular, $ge = e$, so $g =e$. Therefore
$$\ker(\varphi) = \{e\}.$$
The kernel of the left regular action is $\{e\}$. So the left regular action is faithful.
\end{proof}

Exercise 1.7.13 (Kernel of the left regular action)

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Exercise 1.7.13 (Kernel of the left regular action)

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