Exercise · ZW-P-GROUP-007 · ★★☆ Intermediate
A finite group has an element of order \(15\). Can it be a \(p\)-group?
Choose one option.
Show answer
- D. no, because \(15\) does not divide any power of a prime
Explanation
In a \(p\)-group the order of each element divides \(p^n\), so it is a power of \(p\). Since \(15=3\cdot5\) is not a prime power, it cannot happen. Being odd is not the reason: \(3\)-groups have odd order.
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-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.