Exercise · ZW-ORBIT-AND-STABILIZER-005 · ★☆☆ Basic
For an action of \(G\) on \(X\) and \(x\in X\), which statement is always true?
Choose one option.
Show answer
- D. The stabilizer of \(x\) is a subgroup of \(G\)
Explanation
If \(g,h\) fix \(x\), so do \(gh\) and \(g^{-1}\), and \(e\) fixes it: the stabilizer is a subgroup of \(G\). The orbit lives in \(X\) and is not all of \(X\) in general.
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
- ZW-ORBIT-AND-STABILIZER-001 The group S_3 acts on \1,2,3\ by g· x=g(x). How many elements does the orbit of 1 have?
- ZW-ORBIT-AND-STABILIZER-002 Let G=langle(1\ 2\ 3)(4\ 5)rangle≤ S_5 act by g· x=g(x). What is the orbit of 4?
- ZW-ORBIT-AND-STABILIZER-003 With the same group G=langle(1\ 2\ 3)(4\ 5)rangle, what is the stabilizer of 1?
- ZW-ORBIT-AND-STABILIZER-004 The symmetry group of the square (order 8) acts on its four vertices. How many elements does the st…
- ZW-ORBIT-AND-STABILIZER-006 Let G=langle(1\ 2)(3\ 5)rangle≤ S_5 act on \1,2,3,4,5\. How many distinct orbits are there?
- ZW-ORBIT-AND-STABILIZER-007 A group G acts on itself by g· x=gx. What is the stabilizer of an element x?
- ZW-ORBIT-AND-STABILIZER-008 If y=g· x in an action of G, what is the stabilizer of y in terms of that of x?
- ZW-ORBIT-AND-STABILIZER-009 S_4 acts on the two-element subsets of \1,2,3,4\ by g·\a,b\=\g(a),g(b)\. How many elements does the…