Direct Sum of Groups

Let $(G_i)_{i\in I}$ be a family of groups with identities $e_i$. The (external) direct sum

is the subgroup of the direct product $\prod_{i\in I} G_i$ consisting of all tuples $(g_i)_{i\in I}$ such that $g_i=e_i$ for all but finitely many indices $i$ (this “all but finitely many” condition is called finite support).

For finite index sets $I$, the direct sum coincides with the direct product. In practice the term “direct sum” is most common for abelian groups and infinite families, where the finite-support condition matters.

Examples:

• If $I=\{1,\dots,n\}$ is finite, then $\bigoplus_{i=1}^n G_i=\prod_{i=1}^n G_i$ as groups.
• $\bigoplus_{n\in\mathbb{N}}\mathbb{Z}$ is the free abelian group on countably many generators (elements are integer sequences that are eventually $0$).
• $\bigoplus_{n\in\mathbb{N}} C_2$ is the group of $\{0,1\}$-sequences with finitely many $1$’s under componentwise addition mod $2$.