Jorge Martín: Hahn's Embedding Theorem and some Results on Ordered Groups

Abstract:

The theory of ordered groups studies the structure of groups whose underlying set is ordered in such a way that the binary operation is compatible with the order. One fundamental result in this area is Hahn’s embedding theorem, which provides a characterisation of linearly ordered abelian groups by means of an isomorphism with an additive subgroup of some product of the real numbers. In this talk, I will develop the theory of ordered groups necessary to obtain a self-contained proof of this theorem, namely stating Hölder’s theorem on Archimedean groups first. In addition, the introduced concepts will allow us to discuss central results on orderability, showing examples of families of groups that admit a compatible order. Finally, I will comment on some direct applications of Hahn’s theorem, especially its relation to the search for power series solutions of differential equations.