In Chapter 5 we proved the duality theorem for compact groups and discrete groups. To extend the duality theorem to all LCA-groups we will prove two special cases of the following proposition: If G is an LCA-group with a subgroup H such that both H and G/H satisfy the duality theorem, then G satisfies the duality theorem. The two cases we prove are when H is compact and when H is open. The duality theorem for all LCA-groups then follows from the fact that every LCA-group G has an open subgroup H which in turn has a compact subgroup K such that H/K is an “elementary group” which is known to satisfy the duality theorem. By an “elementary group” we mean one which is of the form Ra × Zb × Tc × F, where F is a finite discrete abelian group and a, b and c are non-negative integers. Once we have the duality theorem we use it, together with the above structural result, to prove the Principal Structure Theorem.

We begin with some structure theory.

Definition. A topological group is said to be monothetic if it has a dense cyclic subgroup.

Examples. Z and T are monothetic.

Theorem 19. Let G be a monothetic LCA-group. Then either G is compact or G is topologically isomorphic to Z.

Proof. If G is discrete then either G = Z or G is a finite cyclic group and hence is compact. So we have to prove that G is compact if it is not discrete.