The group direct sum of a sequence {G_n}_(n = 0)^∞ of groups G_n is the set of all sequences {g_n}_(n = 0)^∞, where each g_n is an element of G_n, and g_n is equal to the identity element of G_n for all but a finite set of indices n. It is denoted ⊕_(n = 0)^∞ G_n, and it is a group with respect to the componentwise operation derived from the operations of the groups G_n. This definition can easily be extended to any collection {G_i}_(i element I) of groups, where I is any finite or infinite set of indices.
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