# Can someone explain the precise difference between of direct sum and direct product of groups?

As far as I know, the direct product of groups $G_1, \dots , G_n$ is the group with the underlying set being the cartesian product and the operation done component wise. It's not clear to me what a direct sum of groups $G_1, \dots ,G_n$ really means. Wikipedia makes it sound like the term "direct sum" is used to refer to the direct product of abelian groups.

Some clarification would be appreciated.

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The direct sum of a family $\{G_i:i\in I\}$ of groups is the same as the direct product when $I$ is finite. When $I$ is infinite, however, the direct sum is a proper subgroup of the direct product: it’s the set of $g\in\prod_{i\in I}G_i$ such that $g_i=1_{G_i}$ for all but at most finitely many $i\in I$.
Also, note that the set $\prod_{i\in I}G_i$ is a group whose elements are all vectors $(g_k)$ in the Cartesian product and whose operation is $(g_k)+(g'_k)=(g_k+g'_k)$. – Babak S. Jan 18 '13 at 3:44
@Babak: That is also true of the direct sum. The difference is that in the direct sum all but finitely many of the $g_k$ and $g_k'$ are the identity in $G_k$. – Brian M. Scott Jan 18 '13 at 3:46