Stabilizer of Element of Group Acting on Itself is Trivial

Theorem

Let $\struct {G, \circ}$ be a group whose identity is $e$.

Let $*$ be the group action of $\struct {G, \circ}$ on itself by the rule:

$\forall g, h \in G: g * h = g \circ h$

Then the stabilizer of an element $x \in G$ is given by:

$\Stab x = \set e$

Proof

Let $g \in \Stab x$.

Then:

\(\ds g * x\)

\(=\)

\(\ds x\)

Definition of Stabilizer

\(\ds \leadsto \ \ \)

\(\ds g \circ x\)

\(=\)

\(\ds x\)

Definition of Group Action (this particular one)

\(\ds \leadsto \ \ \)

\(\ds g\)

\(=\)

\(\ds e\)

Definition of Identity Element

Hence the result, by definition of trivial subgroup.

$\blacksquare$

Also see

• Group Acts on Itself
• Orbit of Element of Group Acting on Itself is Group

Sources

• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $10$: The Orbit-Stabiliser Theorem: Example $10.10$