Further Examples
Amongst groups that are not Lie groups, and so do not carry the structure of a manifold, examples are the additive group Zp of p-adic integers, and constructions from it. In fact any profinite group is a compact group. This means that Galois groups are compact groups, a basic fact for the theory of algebraic extensions in the case of infinite degree.
Pontryagin duality provides a large supply of examples of compact commutative groups. These are in duality with abelian discrete groups.
Read more about this topic: Compact Group
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