Exercise · ZW-ORBIT-STABILIZER-THEOREM-010 · ★★★ Advanced
A group of order \(24\) acts transitively on a set of \(6\) elements. If \(x,y\in X\), how many \(g\in G\) satisfy \(g\cdot x=y\)?
Choose one option.
Show answer
- C. \(4\)
Explanation
There is a \(g_0\) with \(g_0\cdot x=y\), and the \(g\) that do it are \(g_0h\) with \(h\in\mathrm{Stab}(x)\). Since \(|\mathrm{Stab}(x)|=24/6=4\), there are \(4\) of them.
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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