Exercise · ZW-ORBIT-STABILIZER-THEOREM-009 · ★★★ Advanced
A group of order \(9\) acts on a set of \(7\) elements. What is the minimum number of fixed points (one-element orbits) there must be?
Choose one option.
Show answer
- B. \(1\)
Explanation
Orbit sizes divide \(9\) and are at most \(7\): they are \(1\) or \(3\). With \(7=3a+b\) and \(b\) fixed points, \(b=7-3a\equiv1\pmod 3\), so \(b\ge1\) (attained with \(a=2\)).
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
- ZW-ORBIT-STABILIZER-THEOREM-001 A group of order 12 acts on a set and the stabilizer of x has 3 elements. How many elements does th…
- ZW-ORBIT-STABILIZER-THEOREM-002 A group of order 20 acts and the orbit of x has 5 elements. How many elements does the stabilizer o…
- ZW-ORBIT-STABILIZER-THEOREM-003 S_4 acts on \1,2,3,4\ by g· x=g(x). How many elements does the stabilizer of 1 have?
- ZW-ORBIT-STABILIZER-THEOREM-004 A group of order 15 acts on a set. Which of these CANNOT be the size of an orbit?
- ZW-ORBIT-STABILIZER-THEOREM-005 S_5 acts on the 3-element subsets of \1,2,3,4,5\, all forming a single orbit. How many elements doe…
- ZW-ORBIT-STABILIZER-THEOREM-006 The rotation group of the cube has order 24 and acts on its 8 vertices, all in one orbit. How many…
- ZW-ORBIT-STABILIZER-THEOREM-007 For an action of a finite group G on X and x∈ X, which statement is true?
- ZW-ORBIT-STABILIZER-THEOREM-008 A group of order 60 acts on X with a single orbit, and the stabilizer of each point has 5 elements.…
← ZW-ORBIT-STABILIZER-THEOREM-008 ZW-ORBIT-STABILIZER-THEOREM-010 →