Exercise · ZW-ORBIT-STABILIZER-THEOREM-005 · ★★☆ Intermediate

\(S_5\) acts on the \(3\)-element subsets of \(\{1,2,3,4,5\}\), all forming a single orbit. How many elements does the stabilizer of \(\{1,2,3\}\) have?

Choose one option.

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  • A. \(12\)

Explanation

There are \(\binom{5}{3}=10\) subsets, so \(|\mathrm{Stab}|=120/10=12\). Directly: \(3!\cdot2!=12\) permutations preserve \(\{1,2,3\}\).

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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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