Exercise · ZW-ORBIT-AND-STABILIZER-010 · ★★★ Advanced
In an action of \(G\) on \(X\), the stabilizer of \(x\) is all of \(G\). What is the orbit of \(x\)?
Choose one option.
Show answer
- A. \(\{x\}\)
Explanation
If every \(g\in G\) satisfies \(g\cdot x=x\), the only element of the orbit is \(x\) itself: \(x\) is a fixed point of the whole action.
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.
\[\mathrm{Orb}(x)=\{g\cdot x : g\in G\},\qquad \mathrm{Stab}(x)=\{g\in G : g\cdot x=x\}\]
More practice problems: orbit and stabilizer
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- ZW-ORBIT-AND-STABILIZER-007 A group G acts on itself by g· x=gx. What is the stabilizer of an element x?
- ZW-ORBIT-AND-STABILIZER-008 If y=g· x in an action of G, what is the stabilizer of y in terms of that of x?