Exercise · ZW-P-GROUP-008 · ★★★ Advanced
A group of order \(25\) acts on a set of \(11\) elements. What is the minimum number of fixed points?
Choose one option.
Show answer
- B. \(1\)
Explanation
Orbit sizes divide \(25\) and are at most \(11\): they are \(1\) or \(5\). With \(a\) orbits of size \(5\) and \(b\) fixed points, \(11=5a+b\) with \(a\le2\), hence \(b\ge1\).
Review the concept: p-group
A \(p\)-group is a finite group \(G\) whose order is a power \(p^n\), with \(n\ge1\), of a prime number \(p\).
\[|G|=p^{n}\]
More practice problems: p-group
- ZW-P-GROUP-001 Which of these groups is a 2-group?
- ZW-P-GROUP-002 In a group of order 125, which cannot be the order of an element?
- ZW-P-GROUP-003 A subgroup of a group of order 16 cannot have order:
- ZW-P-GROUP-004 Is ℤ_9×ℤ_3 a p-group? For which prime p?
- ZW-P-GROUP-005 A group of order 81 has a subgroup of order 9. What is the index of that subgroup?
- ZW-P-GROUP-006 For which primes p is a group of order 12 a p-group?
- ZW-P-GROUP-007 A finite group has an element of order 15. Can it be a p-group?
- ZW-P-GROUP-009 Select the true statement about a finite p-group G.