Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 2005 • Question 1

Question

Math 6310 • Algebra Fall 2005 • Question 1

Let \(G\) be a finite group and \(p\) a prime number such that \(|G|=p^n\) for some \(n>0\). Prove that \(G\) has nontrivial center and that for all integers \(0<m<n\), \(G\) has at least one normal subgroup of order \(p^m\).

Raw LaTeX

Let \(G\) be a finite group and \(p\) a prime number such that \(|G|=p^n\) for some \(n>0\).
Prove that \(G\) has nontrivial center and that for all integers \(0<m<n\), \(G\) has at least one
normal subgroup of order \(p^m\).

Topic Tags

Finite Group Theory Normal Subgroups and Commutator Subgroups p-Groups