conjugate stabilizer subgroups

Let $\mathrm{\cdot }$ be a right group action of $G$ on a set $M$. Then

$$G_{\alpha \cdot g}=g^{-1}{G}_{\alpha }g$$

for any $\alpha \in M$ and $g\in G$.

Proof:

$$x\in G_{\alpha \cdot g}\leftrightarrow \alpha \cdot (gx)=\alpha \cdot g\leftrightarrow \alpha \cdot (gx{g}^{-1})=\alpha \leftrightarrow gx{g}^{-1}\in G_{\alpha }\leftrightarrow x\in g^{-1}\alpha g$$

and therefore $G_{\alpha \cdot g}=g^{-1}{G}_{\alpha }g$.

Thus all stabilizer subgroups for elements of the orbit $G(\alpha )$ of $\alpha$ are conjugate to $G_{\alpha }$.

Title	conjugate stabilizer subgroups
Canonical name	ConjugateStabilizerSubgroups
Date of creation	2013-03-22 13:21:44
Last modified on	2013-03-22 13:21:44
Owner	Thomas Heye (1234)
Last modified by	Thomas Heye (1234)
Numerical id	7
Author	Thomas Heye (1234)
Entry type	Derivation
Classification	msc 20A05
Related topic	Orbit