Exercise · ZW-P-GROUP-004 · ★☆☆ Basic
Is \(\mathbb{Z}_{9}\times\mathbb{Z}_{3}\) a \(p\)-group? For which prime \(p\)?
Choose one option.
Show answer
- D. yes, for \(p=3\)
Explanation
Its order is \(9\cdot3=27=3^3\), a power of the prime \(3\). Not being cyclic does not matter, and \(9\) is not prime.
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-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?
- ZW-P-GROUP-009 Select the true statement about a finite p-group G.