Counting subgroups and topological group topologies

Abstract

Algebra. Either |^(G)| = 2a or |^(G)| = a. If \S?(G) = a then a = co. We describe and characterize those (countable) G such that \S?(G) = ω, and we give several examples. Topology. If γ 2 α, then &(y) = 0; otherwise 2 αγ . If γ > 2 α then Jt(y) = 0; if log(α) < γ < 2 α then = 2 α γ; and if ω < γ < a then K(γ)| = 2 α. 0. Introduction and motivation. As a reading of the Synopsis may suggest, this work originated with the authors' interest in the following questions: Given an infinite Abelian group, how many topological group topologies does G possess? Of these, how many may be chosen pairwise non-homeomorphic? How many metrizable? How many totally bounded? How many totally bounded and metrizable? We approached the latter questions through a result from [6] which gives a one-to-one order-preserving correspondence between the set SS(G) of totally bounded topological group topologies for G and the set of point-separat ing subgroups of the homomorphism group Hom(G, T). Thus it became natural—indeed necessary—to count the number of subgroups of a group of the form Hom(G,T). In §§1 and 2, which we believe have algebraic interest quite independent of their topological roots, we do a bit more: We show that every uncountable Abelian group G has 2|G| subgroups, and we describe in some detail the fine algebraic structure of what we call ω-groups. These are by definition the (necessarily countable) Abelian groups G with fewer then 2|G| subgroups; we show that each ω-group has exactly ω-many subgroups, and we describe the relationship between the ω-groups and the so-called q.d. groups of Beaumont and Pierce [2]. The algebraic analysis of §1, together with the result cited from [6], allows us to describe some gross features of the partially ordered sets £%(G). Here our work is sufficiently coarse that the various cardinal