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 action.

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

$\blacksquare$

Sources

• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $10$: The Orbit-Stabiliser Theorem: Exercise $2$