Exercise · ZW-ORBIT-AND-STABILIZER-003 · ★★☆ Intermediate
With the same group \(G=\langle(1\ 2\ 3)(4\ 5)\rangle\), what is the stabilizer of \(1\)?
Choose one option.
Show answer
- B. \(\{e,(4\ 5)\}\)
Explanation
The generator \(g\) has order \(6\), and \(g^k\) fixes \(1\) when \(3\mid k\). This gives \(g^0=e\) and \(g^3=(4\ 5)\). Also \((1\ 2\ 3)\) moves \(1\), so it is not in the stabilizer.
Review the concept: Orbit and stabilizer
If a group \(G\) acts on a set \(X\), the orbit of \(x\in X\) is the set of all its images \(g\cdot x\), and its stabilizer is the subgroup of elements of \(G\) that leave it fixed.
\[\mathrm{Orb}(x)=\{g\cdot x : g\in G\},\qquad \mathrm{Stab}(x)=\{g\in G : g\cdot x=x\}\]
More practice problems: orbit and stabilizer
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