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For questions about taking the direct sum of groups and other algebraic structures.

Direct sums are defined for a number of different sorts of mathematical objects, including subspaces, matrices, modules, and groups.

Definition: Let $$~U,~ W~$$ be subspaces of $$~V~$$ . Then $$~V~$$ is said to be the direct sum of $$~U~$$ and $$~W~$$, and we write $$~V = U ⊕ W~$$, if $$~V = U + W~$$ and $$~U ∩ W = \{0\}~$$.

• The significant property of the direct sum is that it is the coproduct in the category of modules (i.e., a module direct sum).
• Direct products and direct sums differ for infinite indices. An element of the direct sum is zero for all but a finite number of entries, while an element of the direct product can have all nonzero entries.
• The direct sum of Abelian groups is the same as the group direct product, but that the term direct sum is not used for groups which are non-Abelian.

References:

https://en.wikipedia.org/wiki/Direct_sum

http://mathworld.wolfram.com/DirectSum.html