Exercise · ZW-ORBIT-STABILIZER-THEOREM-001 · ★☆☆ Basic
A group of order \(12\) acts on a set and the stabilizer of \(x\) has \(3\) elements. How many elements does the orbit of \(x\) have?
Choose one option.
Show answer
- B. \(4\)
Explanation
\(|\mathrm{Orb}(x)|=|G|/|\mathrm{Stab}(x)|=12/3=4\). Multiplying, as in \(36\), is the typical mistake.
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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