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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Spring 2011 • Question 3

Question

Math 6310 • Algebra Spring 2011 • Question 3

Let \(G\) be a \(p\)-group with \(|G|>p\). Show that

  1. \(G\) has nontrivial center;
  2. \(G\) has a normal subgroup of every order \(p^m<|G|\).

Raw LaTeX

Let \(G\) be a \(p\)-group with \(|G|>p\). Show that
\begin{enumerate}
\item \(G\) has nontrivial center;
\item \(G\) has a normal subgroup of every order \(p^m<|G|\).
\end{enumerate}

Topic Tags

Finite Group Theory Normal Subgroups and Commutator Subgroups p-Groups