Stabilizer of Subgroup Action is Identity

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Theorem

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

Let $\struct {H, \circ}$ be a subgroup of $G$.

Let $*: H \times G \to G$ be the subgroup action defined for all $h \in H, g \in G$ as:

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


The stabilizer of $x \in G$ is $\set e$:

$\Stab x = \set e$


Proof

From Subgroup Action is Group Action we have that $*$ is a group action.

Let $x \in G$.

Then:

\(\ds \Stab x\) \(=\) \(\ds \set {h \in H: h * x = x}\) Definition of Stabilizer
\(\ds \) \(=\) \(\ds \set {h \in H: h \circ x = x}\) Definition of $*$
\(\ds \) \(=\) \(\ds \set {h \in H: h = x \circ x^{-1} }\)
\(\ds \) \(=\) \(\ds \set {h \in H: h = e}\)
\(\ds \) \(=\) \(\ds \set e\)

Hence the result, by definition of right coset.

$\blacksquare$


Sources

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