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The kernel of a group action is precisely the kernel of the induced permutation representation


Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.7 Exercise 1.7.5

Prove that the kernel of an action of the group $G$ on a set $A$ is the same as the kernel of the corresponding permutation representation $G \rightarrow S_A$.


Solution: Let $G$ be a group acting on $A$. The kernel of the action is the set $$K = \{ g \in G \ |\ g \cdot a = a\ \mathrm{for\ all}\ a \in A \}.$$ The corresponding permutation representation is a group homomorphism $\varphi : G \rightarrow S_A$ given by $\varphi(g)(a) = g \cdot a$, and by definition $$\mathsf{ker}\ \varphi = \{ g \in G \ |\ \varphi(g) = 1 \}.$$ $K \subseteq \mathsf{ker}\ \varphi$: Let $k \in K$. Then for all $a \in A$, we have $$\varphi(k)(a) = k \cdot a = a,$$ so that $$\varphi(k) = \mathsf{id}_A = 1.$$ Thus $g \in \mathsf{ker}\ \varphi$.

$\mathsf{ker}\ \varphi \subseteq K$: Let $k \in \mathsf{ker}\ \varphi$. Then for all $a \in A$, we have $$k \cdot a = \varphi(k)(a) = \mathsf{id}_A(a) = a.$$ Thus $k \in K$.

Linearity

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