Suppose I take an infinite direct power $\prod G$ of some (not necessarily finite) group $G$. I want to know about the subgroups of $\prod G$ that are maximal subject to having trivial intersection with $\oplus G$. Is there a general description of such subgroups (in terms of ultrafilters maybe), without delving into the structure of $G$? If not, are there at least some interesting general constructions of 'large' subgroups of $\prod G$ that intersect trivially with $\oplus G$?
One way to construct subgroups intersecting trivially with $\oplus G$ is as follows: take a family $\mathcal{P}$ of partitions of the indexing set $I$ that is closed under coarsest common refinement, such that all the parts in any given $P \in \mathcal{P}$ are infinite. (For instance $I = \mathbb{Z}$ and $\mathcal{P}$ is congruence classes modulo $n$.) Now take all those $g \in \prod G$ for which there is some $P$ in $\mathcal{P}$ for which the $i$-th entry of $g$ is determined by which part of $P$ contains $i$.
Are the groups I just described ever maximal?
Best Answer
Here is an example showing that the particular subgroups you describe need not be maximal.
Consider the case of $\Pi_{n\in\mathbb{N}} \mathbb{Z}/2\mathbb{Z}$, with the subgroup $H$ consisting of the periodic elements of the product, which I think is the same as the subgroup you describe in your modulo $n$ example. This is a subgroup and has trivial intersection with the direct sum.
But I claim that it is not maximal. The reason is that this $H$ is countable, but I claim any subgroup of this product that is maximal with respect to having trivial intersection with the direct sum must be uncountable (and in fact must have size continuum). To see this, suppose that $H$ is a countable subgroup having trivial intersection with the direct sum. Thus also, the set of elements of the product that differ at most finitely from an element in $H$ is also countable. Since the direct product is uncountable, there is an element $h$ in the direct product that does not differ finitely from any element of $H$. It follows that the group generated by $H$ and $h$ continues to have trivial intersection with the direct sum. So $H$ is not maximal.
More generally, the same argument shows in this case that any maximal group with the property must have size continuum, for if not, then the set of elements differing finitely from $H$ will also have size less than continuum, and we will be able to find an $h$ that can be added.