On the Poset of Totally Dense Subgroups of Compact Groups by W. W. Comfort and Dikran Dikranjan

Electronic Version 24 Summer (1999)

On the Poset of Totally Dense Subgroups of Compact Groups

W. W. Comfort and Dikran Dikranjan

A subgroup H of a topological group G is said to be essential [resp., totally dense] in G if |H \cap N| > 1 [resp., H \cap N is dense in N] for every closed, non-trivial normal subgroup N of G. We study the poset ED(K) of all essentially dense subgroups of a compact (abelian) group K and its subposet TD(K) of totally dense subgroups. Specifically we show:

Theorem A. For a compact abelian group K the following are equivalent:

K admits a smallest totally dense subgroup;

TD(K) is a (complete) lattice;

the torsion subgroup t(K) of K is totally dense in K;

t(K) is essentially dense in K; and

K contains copies of the group Z_p of p-adic integers for no prime p.

Theorem B. For a compact abelian group K the following are equivalent:

K admits a smallest essentially dense subgroup;

ED(K) is a (complete) lattice;

K admits a smallest totally dense subgroup and soc(K) is either dense or open in K.

We study also the class C of topological groups K such that every essentially dense subgroup of K is totally dense, and we describe the compact groups in C that are either abelian or connected.

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