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