Exercise · ZW-ORBIT-AND-STABILIZER-007 · ★★☆ Intermediate
A group \(G\) acts on itself by \(g\cdot x=gx\). What is the stabilizer of an element \(x\)?
Choose one option.
Show answer
- D. \(\{e\}\)
Explanation
If \(gx=x\), multiplying by \(x^{-1}\) gives \(g=e\). Only the identity fixes \(x\).
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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