Exercise · ZW-ORBIT-AND-STABILIZER-001 · ★☆☆ Basic
The group \(S_3\) acts on \(\{1,2,3\}\) by \(g\cdot x=g(x)\). How many elements does the orbit of \(1\) have?
Choose one option.
Show answer
- C. \(3\)
Explanation
For each \(k\in\{1,2,3\}\) some permutation sends \(1\) to \(k\), so the orbit is all of \(\{1,2,3\}\). The \(6\) is the order of \(S_3\), not the size of the orbit.
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-008 If y=g· x in an action of G, what is the stabilizer of y in terms of that of x?
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