Note on Linearly Compact Abelian Groups

Abstract

1By a group A is meant throughout an additively written abelian group. A is said to have a linear topology if there is a system of subgroups U\ (i el) of A such that, for any a e A, the cosets a-\-U i (i el) form a fundamental system of neighborhoods of a. The group operations are continuous in any linear topology; the topologies are always assumed to be Hausdorff, that is, P|,-.U t = 0. A linearly compact group is a group A with a linear topology such that if a,-f^4,-(j el) is a system of cosets modulo closed subgroups Aj with the finite intersection property (i.e. any finite number of a^Aj have a non-void intersection), then the intersection |~) 3 (a^Aj) of all of them is not empty.Our present aim is to study the algebraic structure of linearly compact groups. They can be characterized as inverse limits of groups with minimum condition on subgroups. It turns out that the class of linearly compact groups lies between the classes of compact and algebraically compact groups: every group that admits a compact topology is linearly compact in some suitable topology, while all linearly compact groups are algebraically compact. A complete system of invariants can be obtained for linearly compact groups. In obtaining these invariants, a fundamental role is played by the fact that every linearly compact group A can be decomposed as a direct product A -Y^^Aj, where, for each prime p, A v is a topological module over the />-adic integers, with the product topology on A. For the components A p we can apply the duality theory of Kaplansky [10] and Schoneborn [15] (between discrete and linearly compact />-adic modules) in order to get a full structure theorem on linearly compact groups.We shall use the following notations: Z = infinite cyclic group; Z(n) = cyclic group of finite order n; Z(p°°) = group of type p°°; Q = group of rational numbers; R = group of real numbers; K = real numbers mod 1; J v = group of ^>-adic integers; K v = group of ^>-adic numbers. The symbols © and ]T denote direct sums and products, respectively.

Inverse limits of algebras as retracts of their direct products

“…As each Z/p n Z is finite, J p is linearly, and hence algebraically compact. (See [1] and [2].) Since, as can easily be proved, J p is pure in n Z/p n Z, it follows that the canonical monomorphism 0 → lim ← − Z/p n Z → n Z/p n Z splits.…”