Well-known ${\rm LCA}$ groups characterized by their closed subgroups

Abstract

In this paper we determine (1) the class of all non- discrete LCA groups for which every proper closed subgroup is the kernel of a continuous character of the group, (2) the class of locally compact groups whose closed subgroups are totally ordered by in- clusion, and (3) the class of infinite LCA groups whose proper closed subgroups are topologically isomorphic. Since all these de- terminations involve only the most common LCA groups, we may regard our findings as characterizations of natural classes of these well-known groups. The program of deriving information about locally compact Abe- lian (LCA) groups from a knowledge of their closed subgroups has received attention in recent years; see, for example, references (3), (4), and (5). In this paper we shall state and prove three theorems characterizing some of the most common LCA groups by means of very natural hypotheses upon their closed subgroups. The LCA groups of which we shall make constant use are the circle T, the additive real numbers R, the integers Z, the cyclic groups Z{n), the quasicyclic groups Z(p°°), the £-adic integers JP and the p-adic numbers Fv. Precise definitions of all these groups may be found in (l); in particular, much detailed information on the groups Jp and Fp may be found in (l, §§10 and 25). Except where explicitly stated, all groups throughout are assumed to be LCA and Hausdorff topological groups. If the group Gi is topologically isomorphic to the group G2, we write Gi=G2. The character group of a group G is de- noted by G, and the kernel of a character 7 is written as ker y. Finally, if x is an element of a group G, we use the symbol (x) to denote the subgroup of G generated by x; the closure of this subgroup is written as (x). These findings constitute part of the author's doctoral disserta- tion submitted to Stanford University in 1969. The author is grateful to the National Science Foundation for financial support and to