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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Spring 2017 • Question 4

Question

Math 6310 • Algebra Spring 2017 • Question 4

Let \(G\) be a finite \(p\)-group, for \(p\) a prime integer. Let \(e\) denote the identity element, and \(Z\) the center of \(G\). If \(N\ne \{e\}\) is normal, prove that \(N\cap Z\ne \{e\}\).

Raw LaTeX

Let \(G\) be a finite \(p\)-group, for \(p\) a prime integer. Let \(e\) denote the identity element,
and \(Z\) the center of \(G\). If \(N\ne\{e\}\) is normal, prove that \(N\cap Z\ne\{e\}\).

Topic Tags

Finite Group Theory Normal Subgroups and Commutator Subgroups p-Groups