Exercise · ZW-ORBIT-AND-STABILIZER-008 · ★★☆ Intermediate
If \(y=g\cdot x\) in an action of \(G\), what is the stabilizer of \(y\) in terms of that of \(x\)?
Choose one option.
Show answer
- A. \(g\,\mathrm{Stab}(x)\,g^{-1}\)
Explanation
If \(h\cdot x=x\), then \((ghg^{-1})\cdot y=g\cdot(h\cdot x)=g\cdot x=y\). This gives one inclusion, and using \(g^{-1}\) the other: \(\mathrm{Stab}(y)=g\,\mathrm{Stab}(x)\,g^{-1}\). Points in one orbit have conjugate stabilizers, not necessarily equal ones.
Review the concept: Orbit and stabilizer
If a group \(G\) acts on a set \(X\), the orbit of \(x\in X\) is the set of all its images \(g\cdot x\), and its stabilizer is the subgroup of elements of \(G\) that leave it fixed.
\[\mathrm{Orb}(x)=\{g\cdot x : g\in G\},\qquad \mathrm{Stab}(x)=\{g\in G : g\cdot x=x\}\]
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