Exercise · ZW-ORBIT-STABILIZER-THEOREM-007 · ★★☆ Intermediate

For an action of a finite group \(G\) on \(X\) and \(x\in X\), which statement is true?

Choose one option.

Show answer
  • C. \(|\mathrm{Orb}(x)|\) divides \(|G|\)

Explanation

From \(|\mathrm{Orb}(x)|\cdot|\mathrm{Stab}(x)|=|G|\) it follows that \(|\mathrm{Orb}(x)|\) divides \(|G|\). The others fail: in the action of \(G\) on itself the orbit has \(|G|\) elements and the stabilizer only one.

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Review the concept: Orbit-stabilizer theorem

For a finite group \(G\) acting on \(X\) and an element \(x\in X\), the size of the orbit of \(x\) is the index of its stabilizer in \(G\).

\[|\mathrm{Orb}(x)|=[G:\mathrm{Stab}(x)]=\frac{|G|}{|\mathrm{Stab}(x)|}\]

More practice problems: orbit-stabilizer theorem

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