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.
Best Answer
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).