# 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.

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 $(g_k)+(g'_k)=(g_k+g'_k)$. – Mikasa 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