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

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.


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.

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