Exercise · ZW-ORBIT-AND-STABILIZER-002 · ★☆☆ Basic
Let \(G=\langle(1\ 2\ 3)(4\ 5)\rangle\le S_5\) act by \(g\cdot x=g(x)\). What is the orbit of \(4\)?
Choose one option.
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- B. \(\{4,5\}\)
Explanation
Powers of the generator send \(4\) to \(5\) and \(5\) to \(4\), and never mix them with \(1,2,3\). The orbit is \(\{4,5\}\).
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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