Group Actions on Sets

Take a group $(G, \cdot)$ , and a set $X$ . Assume we have $μ : G \times X \to X$ .

Denote $μ (g, x) = g ∙ x$ the “action” of $g \in G$ on $x \in X$ such that:

$G L_{2} (R)$ act on $R^{2}$ by $A ∙ v = A v$ ← $2 \times 2$ matrix and $(x y)$ column vector. $A ∙ v = (a_{11} x + a_{12} y a_{21} x + a_{22} y) \leftarrow R^{2}$
$(S_{n}, \cdot)$ symmetric permutation group acts on ${1, 2, \dots, n}$ by $σ \cdot i = σ (i)$ where $σ = (1 σ (1) 2 σ (2) \dots \dots n σ (n))$
$(G, \cdot)$ group. $G$ acts on itself by left multiplication: $g ∙ g^{'} = g g^{'}$ (note that $∙$ is the action and $g g^{'}$ is the product in $G$ )
$(G, \cdot)$ also acts on $G$ by conjugation: $g ∙ g^{'} = d e f g g^{'} g^{- 1}$ ← conjugation in $G$ . Verification: $μ (g, g^{'}) = g g^{'} g^{- 1} \in G$ , and $μ : G \times G \to G$ is well-defined. $e_{G} ∙ g^{'} = e g^{'} e^{- 1} = g^{'} \forall g^{'} \in G$ $g ∙ (g^{'} ∙ g^{''}) = g ∙ (g^{'} g^{''} (g^{'})^{- 1} = g g^{'} g^{''} (g^{'})^{- 1} (g^{''})^{- 1} = (g g^{'}) g^{''} (g g^{'})^{- 1} = (g g^{'}) ∙ g^{''}$ .

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