Exercise · ZW-ORBIT-AND-STABILIZER-006 · ★★☆ Intermediate
Let \(G=\langle(1\ 2)(3\ 5)\rangle\le S_5\) act on \(\{1,2,3,4,5\}\). How many distinct orbits are there?
Choose one option.
Show answer
- C. \(3\)
Explanation
The orbits are \(\{1,2\}\), \(\{3,5\}\) and \(\{4\}\); they partition \(X\) into \(3\) orbits.
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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