It is a theorem of W. W. Comfort and K. A. Ross that if 𝐺 is a subgroup of a compact Abelian group and 𝑆 denotes the continuous homomorphisms from 𝐺 to the one-dimensional torus, then the topology on 𝐺 is the initial topology given by 𝑆. Assume that 𝐻 is a subgroup of 𝐺. We study how the choice of 𝑆 affects the topological placement and properties of 𝐻 in 𝐺. Among other results, we have made significant progress toward the solution of the following specific questions. How many totally bounded group topologies does 𝐺 admit such that 𝐻 is a closed (dense) subgroup? If C S denotes the poset of all subgroups of 𝐺 that are 𝑆-closed, ordered by inclusion, does C S have a greatest (resp. smallest) element? We say that a totally bounded (topological, resp.) group is an SC group (topologically simple, resp.) if all its subgroups are closed (if 𝐺 and { e } are its only possible closed normal subgroups, resp.) In addition, we investigate the following questions. How many SC-(topologically simple totally bounded, resp.) group topologies does an arbitrary Abelian group 𝐺 admit?

Here are three recently-established theorems from the literature. (A) (2006) Every non-metrizable compact abelian group K has 2^|K| -many proper dense pseudocompact subgroups. (B) (2003) Every non-metrizable compact abelian group K admits 22^|K| -many strictly finer pseudocompact topological group refinements. (C) (2007) Every non-metrizable pseudocompact abelian group has a proper dense pseudocompact subgroup and a strictly finer pseudocompact topological group refinement. (Theorems (A), (B) and (C) become false if the non-metrizable hypothesis is omitted.) With a detailed view toward the relevant literature, the present authors ask: What happens to (A), (B), (C) and to similar known facts about pseudocompact abelian groups if the abelian hypothesis is omitted? Are the resulting statements true, false, true under certain natural additional hypotheses, etc.? Several new results responding in part to these questions are given, and several specific additional questions are posed.

It is a Theorem of W.~ W. Comfort and K.~ A. Ross that if $G$ is a subgroup of a compact Abelian group, and $S$ denotes those continuous homomorphisms from $G$ to the one-dimensional torus, then the topology on $G$ is the initial topology given by $S$. {Assume that $H$ is a subgroup of $G$. We study how} the choice of $S$ affects the topological placement and properties of $H$ in $G$. Among other results, we have {made significant} progress toward the solution of the following specific questions: How many totally bounded group topologies does $G$ admit such that $H$ is a closed (dense) subgroup? If $C_S$ denotes the poset of all subgroups of $G$ that are $S$-closed, ordered by inclusion, does $C_S$ has a greatest (resp. smallest) element? We say that a totally bounded (topological, resp.) group is an \textit{SC-group} (\textit{topologically simple}, resp.) if all its subgroups are closed (if $G$ and $\{e\}$ are its only possible closed normal subgroups, resp.) {In addition, we investigate the following questions.} How many SC-(topologically simple totally bounded, resp.) group topologies does an arbitrary Abelian group $G$ admit?