Stabilizer of Element after Group Action

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Let $\struct {G, \circ}$ be a group.

Let $S$ be a set.

Let $*_S: G \times S \to S$ be a group actions.

Let $x \in S, a \in G$.


$\Stab {a * x} = a^{-1} \circ \Stab x \circ a$


\(\ds \Stab {a * x}\) \(=\) \(\ds \set {g \in G: g * \paren {a * x} = a * x}\) Definition of Stabilizer
\(\ds \) \(=\) \(\ds \set {g \in G: \paren {g \circ a} * x = a * x}\) Group Action Axiom $\text {GA} 2$
\(\ds \) \(=\) \(\ds \set {g \in G: a^{-1} * \paren {g \circ a} * x = a^{-1} * \paren {a * x} }\)
\(\ds \) \(=\) \(\ds \set {g \in G: \paren {a^{-1} \circ \paren {g \circ a} } * x = \paren {a^{-1} \circ a} * x}\) Group Action Axiom $\text {GA} 2$
\(\ds \) \(=\) \(\ds \set {g \in G: \paren {a^{-1} \circ g \circ a} * x = x}\) Group Axiom $\text G 1$: Associativity and Group Axiom $\text G 3$: Existence of Inverse Element
\(\ds \) \(=\) \(\ds a^{-1} \circ \set {g \in G: g * x = x} \circ a\) Definition of Subset Product
\(\ds \) \(=\) \(\ds a^{-1} \circ \Stab x \circ a\) Definition of Stabilizer