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April 27, 2006 Introduction to Compact Groups Reminder: Our website is "http://www.mathematik.tu-darmstadt.de/ lehrmaterial/SS2006/CompGroups/" If {Aj

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April 27, 2006

Introduction to Compact Groups Reminder: Our website is

"http://www.mathematik.tu-darmstadt.de/

lehrmaterial/SS2006/CompGroups/"

If {Aj : j ∈ J} is an arbitrarly family of abelian groups and B is a compact abelian group, then the morphism

Φ: Y

j∈J

Hom(Aj, B) → Hom(M

j∈J

Aj, B)

given by Φ((fj)j∈J)((aj)j∈J) = P

j∈J fj(aj) is an isomorphism of compact abelian groups.

In particular, taking B = T we see that the char- acter group of an arbitrary direct sum of abelian groups is the product of the character groups.

For instance (ZJ)b = TJ . Program for today.

Prove that for any abelian group A and any nonzero a ∈ A there is a character χ ∈ Ab such that χ(a) 6= 0 .

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