Stabilizer is Subgroup/Corollary
Jump to navigation
Jump to search
Corollary to Stabilizer is Subgroup
Let $G$ be a group whose identity is $e$.
Let $G$ act on a set $X$.
Let $x \in X$.
Then:
- $\forall g, h \in G: g * x = h * x \iff g^{-1} h \in \Stab x$
Proof
\(\ds h * x\) | \(=\) | \(\ds g * x\) | ||||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds g^{-1} * \paren {h * x}\) | \(=\) | \(\ds g^{-1} * \paren {g * x}\) | |||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \paren {g^{-1} h} * x\) | \(=\) | \(\ds \paren {g^{-1} g} * x\) | |||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \paren {g^{-1} h} * x\) | \(=\) | \(\ds e * x\) | |||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \paren {g^{-1} h} * x\) | \(=\) | \(\ds x\) | |||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds g^{-1} h\) | \(\in\) | \(\ds \Stab x\) |
$\blacksquare$
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 6.5$. Orbits: Lemma $\text {(ii)}$