# conjugate stabilizer subgroups

Let $\operatorname{\cdot}$ 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 $g\in G$. 11$G_{\alpha}$ is the stabilizer subgroup of $\alpha\in M$.

Proof:

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

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 ConjugateStabilizerSubgroups 2013-03-22 13:21:44 2013-03-22 13:21:44 Thomas Heye (1234) Thomas Heye (1234) 7 Thomas Heye (1234) Derivation msc 20A05 Orbit