The group direct sum of a sequence of groups is the set of all sequences , where each is an element of , and is equal to the identity
element of
for all but a finite set of indices . It is denoted

(1)

and it is a group with respect to the componentwise operation derived from the operations of the groups .

This definition can easily be extended to any collection of groups, where is any finite or infinite set of indices.

If the additional condition on the identity elements is dropped, we get the definition of the group direct product. Hence, the two
notions coincide whenever the set of indices is finite. Thus, for any groups and ,

(2)

denote the same object.

If
and
are subgroups of the same additive group , the equality

(3)

conventionally means that every has a unique decomposition , where and , so that is essentially the same as the set of all ordered pairs .