The Structure of LCA-Groups

In this chapter we will apply the duality theorem for proving structure theorems for LCA groups. As main result we will show that all such groups are isomorphic to groups of the form $\mathbb{R}^n \times H$ for some $n \in \mathbb{N}_0$, such that H is a locally compact abelian group that contains an open compact subgroup K. This theorem will imply better structure theorems if more information on the group is available. For instance it will follow that every compactly generated LCA group is isomorphic to a group of the form $\mathbb{R}^n \times \mathbb{Z}^m\times K$ for some compact group K and some $n, m \in \mathbb{N}_0$, and every compactly generated group of Lie type is isomorphic to one of the form $\mathbb{R}^n \times\mathbb{Z}^m\times \mathbb{T}^l\times F$, for some finite group F and some nonnegative integers n, m and l. To prepare the proofs of these theorems, we start with a section on connectedness in locally compact groups. The main result in that section shows that every totally disconnected locally compact group G has a unit neighborhood base consisting of compact open subgroups of G. The structure theorems will be shown in the second section of this chapter.