Direct sum of groups
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Algebraic structure → Group theory Group theory 


Modular groups

Infinite dimensional Lie group

In mathematics, a group G is called the direct sum ^{[1]}^{[2]} of two subgroups H_{1} and H_{2} if
 each H_{1} and H_{2} are normal subgroups of G
 the subgroups H_{1} and H_{2} have trivial intersection (i.e., having only the identity element in common), and
 G = <H_{1}, H_{2}>; in other words, G is generated by the subgroups H_{1} and H_{2}.
More generally, G is called the direct sum of a finite set of subgroups {H_{i}} if
 each H_{i} is a normal subgroup of G
 each H_{i} has trivial intersection with the subgroup <{H_{j} : j not equal to i}>, 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
 g = h_{1}*h_{2}* ... * h_{i} * ... * h_{n}
 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
 g = (h_{1},h_{2}, ..., h_{i}, ..., h_{n})
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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Direct summand
Given a group , we say that a subgroup is a direct summand of (or that splits form ) if and only if there exist another subgroup such that is the direct sum of the subgroups and
In abelian groups, if is a divisible subgroup of then is a direct summand of .
Examples
 If we take
 it is clear that is the direct product of the subgroups .
 If is a divisible subgroup of an abelian group . Then there exist another subgroup such that
 If is also a vector space then can be written as a direct sum of and another subespace that will be isomorphic to the quotient .
Equivalence of decompositions into direct sums
In the decomposition of a finite group into a direct sum of indecomposable subgroups the embedding of the subgroups is not unique; for example, in the Klein group, V_{4} = C_{2} × C_{2}, we have that
 V_{4} = <(0,1)> + <(1,0)> and
 V_{4} = <(1,1)> + <(1,0)>.
However, it is the content of the RemakKrullSchmidt theorem that given a finite group G = ∑A_{i} = ∑B_{j}, where each A_{i} and each B_{j} is nontrivial and indecomposable, the two sums have equal terms up to reordering and isomorphism.
The RemakKrullSchmidt theorem fails for infinite groups; so in the case of infinite G = H + K = L + M, even when all subgroups are nontrivial and indecomposable, we cannot then assume that H is isomorphic to either L or M.
Generalization to sums over infinite sets
To describe the above properties in the case where G is the direct sum of an infinite (perhaps uncountable) set of subgroups, more care is needed.
If g is an element of the cartesian product ∏{H_{i}} of a set of groups, let g_{i} be the ith element of g in the product. The external direct sum of a set of groups {H_{i}} (written as ∑_{E}{H_{i}}) is the subset of ∏{H_{i}}, where, for each element g of ∑_{E}{H_{i}}, g_{i} is the identity for all but a finite number of g_{i} (equivalently, only a finite number of g_{i} are not the identity). The group operation in the external direct sum is pointwise multiplication, as in the usual direct product.
This subset does indeed form a group; and for a finite set of groups H_{i}, the external direct sum is identical to the direct product.
If G = ∑H_{i}, then G is isomorphic to ∑_{E}{H_{i}}. Thus, in a sense, the direct sum is an "internal" external direct sum. For each element g in G, there is a unique finite set S and unique {h_{i} in H_{i} : i in S} such that g = ∏ {h_{i} : i in S}.