Exercise · ZW-ORBIT-STABILIZER-THEOREM-004 · ★★☆ Intermediate

A group of order \(15\) acts on a set. Which of these CANNOT be the size of an orbit?

Choose one option.

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Explanation

The size of an orbit is \(|G|/|\mathrm{Stab}(x)|\), so it divides \(|G|=15\). Since \(6\) does not divide \(15\), it cannot be an orbit size.

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 \(G\).

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