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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Spring 2014 • Question 1

Question

Math 6310 • Algebra Spring 2014 • Question 1

Let \(G\) be a group of order \(p^3\), where \(p\) is a prime. Suppose \(G\) is not abelian. Show that the center \(Z\) of \(G\) is isomorphic to \(\mathbb {Z}/(p)\) and that \(G/Z\cong \mathbb {Z}/(p)\times \mathbb {Z}/(p)\).

Raw LaTeX

Let \(G\) be a group of order \(p^3\), where \(p\) is a prime. Suppose \(G\) is \emph{not} abelian.
Show that the center \(Z\) of \(G\) is isomorphic to \(\mathbb{Z}/(p)\) and that
\(G/Z\cong\mathbb{Z}/(p)\times\mathbb{Z}/(p)\).

Topic Tags

Abelian Groups and Classification of Finite Abelian Groups Abelian Groups and Direct Sum Decompositions Ideals and Quotient Rings Normal Subgroups and Commutator Subgroups