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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 2000 • Question 8

Question

Math 6310 • Algebra Fall 2000 • Question 8

Let \(G\) be a nonabelian group of order \(p^3\) where \(p\) is a prime. Prove that \(Z(G)\) and \(G'\) are equal and have order \(p\). [Here \(Z(G)\) is the center of \(G\) and \(G'\) is the commutator subgroup of \(G\), i.e., \(G'\) is generated by all elements of the form \(x^{-1}y^{-1}xy\) with \(x,y\in G\).]

Raw LaTeX

Let \(G\) be a nonabelian group of order \(p^3\) where \(p\) is a prime. Prove that \(Z(G)\) and \(G'\)
are equal and have order \(p\). [Here \(Z(G)\) is the center of \(G\) and \(G'\) is the commutator
subgroup of \(G\), i.e., \(G'\) is generated by all elements of the form \(x^{-1}y^{-1}xy\) with
\(x,y\in G\).]

Topic Tags

Finite Group Theory Normal Subgroups and Commutator Subgroups Sylow Theorems p-Groups