Discussion
Which of the following is true ?
Answer: A
The true statement is that a finite group of order \(p^{k}\), \(k \ge 1\), has a non-trivial center.
This is the class equation argument. Write \(|G| = |Z(G)| + \sum [G : C(x_{i})]\), the sum over representatives of the non-central conjugacy classes. Each index \([G : C(x_{i})]\) is a power of \(p\) greater than 1, so \(p\) divides the sum, and \(p\) divides \(|G|\). Hence \(p\) divides \(|Z(G)|\), so the center has at least \(p\) elements.
The other statements are false. A group of order \(p^{2}\) is abelian with proper nontrivial subgroups, so it is not simple. A group of order \(pq\) has a normal Sylow subgroup (for \(p \lt q\), the Sylow \(q\)-subgroup is normal), so it is not simple. A group of order 45 = 9 × 5 has exactly one Sylow 5-subgroup (the count divides 9 and is 1 mod 5), which is normal.
Post your comment
Please log in or register to post a comment.
Examination Categories
- SSC Exams
- Banking Exams
- Teaching Exams
- Civil Services Exams
- Railways Exams
- Engineering Recruitment Exams
- Defence Exams
- State Government Exams
- Police Exams
- Insurance Exams
- Nursing Exams
- Other Government Exams
- NRA CET
- PG Entrance Exams
- Campus Placement
- MBA Entrance Exams
- CUET and UG Entrance
- Judiciary Exams
- Regulatory Body Exams
- Software Development