Exercise 1.7.5 (Kernel of the action)

Question

Prove that the kernel of an action of the of the group $G$ on the set $A$ is the same as the kernel of the corresponding permutation representation $G 	o S_A$ (cf. Exercise 14 in Section 6).

Richard Ganaye · Accepted Answer

\begin{proof} I repeat the argument already given in Exercise 4:

\bigskip
 By definition (see Example 1 p. 43), the kernel of the action is the set
$$K = \{g \in G \mid \forall a \in A,\ g\cdot a = a \}.$$
Consider the map
$$
\varphi
\left\{
\begin{array}{ccl}
G & 	o &S_A\
g & \mapsto &\varphi_g,
\end{array}
ight.
\qquad 	ext{where} \qquad \varphi_g
\left\{
\begin{array}{ccl}
A & 	o &A\
a & \mapsto &\varphi_g(a) = g\cdot a.
\end{array}
ight.
$$
It is proved in the text (p. 42) that $\varphi$ is a homomorphism.

Note that $K = \ker(\varphi)$: for all $g \in G$,
\begin{align*}
g \in \ker(\varphi) & \iff  \varphi(g) = \mathrm{Id}_A\
&\iff \forall a \in A,\ g\cdot a = a\
&\iff g \in K.
\end{align*}
Therefore
$$K = \ker(\varphi).$$

\end{proof}

Exercise 1.7.5 (Kernel of the action)

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Exercise 1.7.5 (Kernel of the action)

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