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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Spring 2006 • Question 1

Question

Math 6310 • Algebra Spring 2006 • Question 1

Let \(G\) be a finite non-abelian group of order \(p^3\) where \(p\) is a prime number. If \(Z\) is the center of \(G\), show that \(G/Z\) is isomorphic to \(\mathbb {Z}/p\mathbb {Z}\oplus \mathbb {Z}/p\mathbb {Z}\).

Raw LaTeX

Let \(G\) be a finite non-abelian group of order \(p^3\) where \(p\) is a prime number. If \(Z\) is the
center of \(G\), show that \(G/Z\) is isomorphic to
\(\mathbb{Z}/p\mathbb{Z}\oplus\mathbb{Z}/p\mathbb{Z}\).

Topic Tags

Abelian Groups and Classification of Finite Abelian Groups Abelian Groups and Direct Sum Decompositions Finite Group Theory Normal Subgroups and Commutator Subgroups