# Stabilizer of Element after Group Action

## Theorem

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

Let $S$ be a set.

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

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

Then:

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

## Proof

 $\ds \Stab {a * x}$ $=$ $\ds \set {g \in G: g * \paren {a * x} = a * x}$ 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

$\blacksquare$