Discrete Abelian Groups and Their Character Groups

Abstract

1. In this paper we construct the character group C of an arbitrary, discrete, abelian group G. We then discuss the geometrical properties of C, and show that C may always be identified with a closed subgroup of a toral group (defined in ?3). Next, we prove the theorem that every closed subgroup C of a toral group may be identified with the character group of some discrete, abelian group G and that G may, in turn, be identified with the group of all continuous characters of C. Finally we show how C and G may be formalized in a new way which seems quite suggestive of the structure of these groups. The theorem on the dual relations between discrete groups G and closed subgroups C of toral groups is a partial extension of Pontrjagin's relation of duality' between countable abelian groups and their character groups. However, it is an essentially simpler theorem, which can be proved without the Weyl-Peter integration process for constructing the characters of C. It would be interesting to prove directly that every bicompact, topological abelian group may be identified with a closed subgroup of a toral group. We would then have an elementary proof of Pontrjagin's relation in its extended form. Our general method of procedure is based on the possibility of describing the groups G and C by certain systems of congruences derived by one of the present authors in a previous paper.2 As regards notation, since we shall be dealing exclusively with abelian groups, we shall represent the group operation according to which the elements of a group are combined by addition. Then every finite linear combination of elements with integer coefficients will have an obvious meaning and will represent a definite group element. When we are dealing with toral groups we shall see that certain infinite combinations will also have a definite meaning, and that the coefficients need not be restricted to integer values.