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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Spring 2004 • Question 1

Question

Math 6310 • Algebra Spring 2004 • Question 1

Suppose \(G\) is a finite group of order a power of a prime. If \(H\) is a nonidentity normal subgroup of \(G\), prove that \(H\) contains a nonidentity element of \(Z(G)\). \[ Z(G)=\{x\in G\mid xy=yx\text { for all }y\in G\}. \]

Raw LaTeX

Suppose \(G\) is a finite group of order a power of a prime. If \(H\) is a nonidentity normal
subgroup of \(G\), prove that \(H\) contains a nonidentity element of \(Z(G)\).
\[
Z(G)=\{x\in G\mid xy=yx\text{ for all }y\in G\}.
\]

Topic Tags

Finite Group Theory Normal Subgroups and Commutator Subgroups Sylow Theorems