# I'm missing something here. Let X = <mo fence="false" stretchy="false">{ <mo stretchy="

I'm missing something here. Let $X=\left\{\left(123\right),\left(132\right),\left(124\right),\left(142\right),\left(134\right),\left(143\right),\left(234\right),\left(243\right)\right\}$, ${A}_{4}$ act on $X$ by conjugation (inner automorphisms) and $x=\left(123\right)$, then $4=|\mathcal{O}\left(x\right)|=|G|/|{G}_{x}|=12/|{G}_{x}|$. However, ${G}_{x}=\left\{1\right\}$
What's wrong here?
You can still ask an expert for help

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

Litzy Fuentes
The stabilizer is not the identity, because your action in conjugation, not multiplication. Every group element stabilizes itself under conjugation, and every power of a group element stabilizes the original element, too.