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
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 5.6$. Stabilizers: Example $111$
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: The Sylow Theorems: $\S 54 \alpha$