Is there a nice way to express the quotient of the direct product of a family of groups with the corresponding direct sum?

Question

Let $I$ be an arbitrary (finite or infinite) set of indices and $\{G_i\}_{i\in I}$ a family of groups indexed by $I$.

The direct product of the family, denoted by $\prod_{i\in I}G_i$, is the set of all the sequences $\{g_i\}_{i\in I}$, where each $g_i$ is an element of $G_i$, with the group operation defined elementwise. The direct sum of the family, denoted by $\bigoplus_{i\in I} G_i$, is the set of all the sequences $\{g_i\}_{i\in I}$ in the direct product, such that the set $\{i\in I|g_i\neq e_i\}$ is finite, where $e_i$ denotes the identity element of $G_i$.

It is easy to check that $\oplus G_i$ is a normal subgroup of $\Pi G_i$.

My question is: is there a nice way to express the quotient $\frac{\Pi G_i}{\oplus G_i}$?

More precisely, is it isomorphic to some other group that can be described more easily?

The only thing I found so far is that two sequences in the direct product are in the same equivalence class if and only if they differ by a finite number of elements.

Answer 1

Even in the case where $I$ is countably infinite and each $G_i$ is the additive group of integers, this quotient is quite complicated. If I remember correctly, it's the product of (1) a rational vector space of dimension $2^{\aleph_0}$ (considered as an additive group) and (2) $\aleph_0$ copies of the additive group of $p$-adic integers for all primes $p$. This is a result of Balcerzyk; here's the MathSciNet reference:

MR0108529 (21 #7245) Balcerzyk, S. On factor groups of some subgroups of a complete direct sum of infinite cyclic groups. (Russian summary) Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys. 7 1959 141–142. (unbound insert).

Answer 2

In https://arxiv.org/abs/1901.05065, §3.D.4, I call this near product $\prod^\star G_i$, and give some references.

"Is it isomorphic to some other group that can be described more easily?": I don't think so.

Near products can be used to define the notion of near wreath product, which naturally appear to describe centralizers in groups such as the quotient of the symmetric group (on an infinite set) by its subgroup of finitely supported permutations.

Independently, a countable group is LEF iff it embeds into a (countable) near product of finite groups.