Exercise · ZW-P-GROUP-009 · ★★★ Advanced
Select the true statement about a finite \(p\)-group \(G\).
Choose one option.
Show answer
- C. every subgroup of \(G\) is a \(p\)-group
Explanation
The order of a subgroup divides \(p^n\), so it is a power of \(p\). The others are false: the symmetry group of the square has order \(8\) and is neither cyclic nor abelian, and has several subgroups of order \(2\).
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-008 A group of order 25 acts on a set of 11 elements. What is the minimum number of fixed points?