PlanetMath: conjugate stabilizer subgroups

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conjugate stabilizer subgroups

(Derivation)

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

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

for any $\alpha \in M$ and $g \in G$ . ¹

Proof:

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

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 $G_{\alpha}$ .

Footnotes

.... ¹: $G_{\alpha}$ is the stabilizer subgroup of $\alpha \in M$ .

"conjugate stabilizer subgroups" is owned by Thomas Heye.

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