In mathematics, a group G is called the direct sum of a set of subgroups {H_{i}} if
 each H_{i} is a normal subgroup of G
 each distinct pair of subgroups has trivial intersection, and
 G = <{H_{i}}>; in other words, G is generated by the subgroups {H_{i}}.
If G is the direct sum of subgroups H and K, then we write G = H + K; if G is the direct sum of a set of subgroups {H_{i}}, we often write G = ∑H_{i}. Loosely speaking, a direct sum is isomorphic to a weak direct product of subgroups.
In abstract algebra, this method of construction can be generalized to direct sums of vector spaces, modules, and other structures; see the article direct sum of modules for more information.
This notation is commutative; so that in the case of the direct sum of two subgroups, G = H + K = K + H. It is also associative in the sense that if G = H + K, and K = L + M, then G = H + (L + M) = H + L + M.
A group which can be expressed as a direct sum of nontrivial subgroups is called decomposable; otherwise it is called indecomposable.
If G = H + K, then it can be proven that:
 for all h in H, k in K, we have that h*k = k*h
 for all g in G, there exists unique h in H, k in K such that g = h*k
 There is a cancellation of the sum in a quotient; so that (H + K)/K is isomorphic to H
The above assertions can be generalized to the case of G = ∑H_{i}, where {H_{i}} is a finite set of subgroups.
 if i ≠ j, then for all h_{i} in H_{i}, h_{j} in H_{j}, we have that h_{i} * h_{j} = h_{j} * h_{i}
 for each g in G, there unique set of {h_{i} in H_{i}} such that
 There is a cancellation of the sum in a quotient; so that ((∑H_{i}) + K)/K is isomorphic to ∑H_{i}
Note the similarity with the direct product, where each g can be expressed uniquely as
Since h_{i} * h_{j} = h_{j} * h_{i} for all i ≠ j, it follows that multiplication of elements in a direct sum is isomorphic to multiplication of the corresponding elements in the direct product; thus for finite sets of subgroups, ∑H_{i} is isomorphic to the direct product ×{H_{i}}.
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Equivalence of direct sums
The direct sum is not unique for a group; for example, in the Klein group, V_{4} = C_{2} × C_{2}, we have that
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