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April 28, 2006 Introduction to Compact Groups An abelian group

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April 28, 2006

Introduction to Compact Groups

An abelian group G is said to be divisible iff (∀g ∈ G)(∀n ∈ N)(∃x ∈ G)n·x = g. An abelian group G is said to be injective if for each pair of abelian group A ⊆ B and each morphism φ:A → G there is a morphism Φ:B → G such that Φ|A = φ.

Theorem. An abelian group G is divisible if and only if it is injective.

Every abelian group is contained in a divisible one. If a subgroup of an abelian group is divisible, then it is a direct summand.

All rational vector group are divisible, all Pr¨ufer groups are divisible. Quotients of divisible groups are divisible.

Corollary. On an abelian group A the characters, i.e. the elements of Ab separate the points.

Program for today.

Characters of compact groups. Biduals Calculation of Tb Exercise session.

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