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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 2012 • Question 2

Question

Math 6310 • Algebra Fall 2012 • Question 2

Let \(S^1\) be the circle group, i.e., the group of all complex numbers of norm 1 with multiplication. If \(A\) is a finite abelian group, define the dual group \(\widehat {A}\) to be the multiplicative group of all group homomorphisms \(A\to S^1\). Prove that \(A\cong \widehat {A}\).

Raw LaTeX

Let \(S^1\) be the circle group, i.e., the group of all complex numbers of norm 1 with
multiplication. If \(A\) is a finite abelian group, define the dual group \(\widehat{A}\) to be the
multiplicative group of all group homomorphisms \(A\to S^1\). Prove that \(A\cong\widehat{A}\).

Topic Tags

Abelian Groups and Classification of Finite Abelian Groups Abelian Groups and Direct Sum Decompositions Finite Group Theory Hom Functors and Module Homomorphisms