each Hi has trivial intersection with the subgroup <>,
G = <>; in other words, G is generated by the subgroups.
If G is the direct sum of subgroups H and K then we write G = H + K, and if G is the direct sum of a set of subgroups then we often write G = ∑Hi. 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 direct sum is commutativeup to isomorphism. That is, if G = H + K then also G = K + H and thus H + K = K + H. It is also associative in the sense that if G = H + K, and K = L + M, then G = H + = H + L + M. A group which can be expressed as a direct sum of non-trivial subgroups is called decomposable, and if a group cannot be expressed as such a direct sum then 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 /K is isomorphic to H
The above assertions can be generalized to the case of G = ∑Hi, where is a finite set of subgroups:
if i ≠ j, then for all hi in Hi, hj in Hj, we have that hi*hj = hj*hi
for each g in G, there exists a unique set of elements hi in Hi such that
There is a cancellation of the sum in a quotient; so that /K is isomorphic to ∑Hi
Note the similarity with the direct product, where each g can be expressed uniquely as Since hi*hj = hj*hi 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, ∑Hi is isomorphic to the direct product ×.
Direct summand
Given a group, we say that a subgroup is a direct summand of if there exists another subgroup of such that. 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 exists another subgroup of such that.
If also has a vector space structure then can be written as a direct sum of and another subspace 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 we have that However, the Remak-Krull-Schmidt theorem states that given a finite group G = ∑Ai = ∑Bj, where each Ai and each Bj is non-trivial and indecomposable, the two sums have equal terms up to reordering and isomorphism. The Remak-Krull-Schmidt theorem fails for infinite groups; so in the case of infinite G = H + K = L + M, even when all subgroups are non-trivial and indecomposable, we cannot conclude that H is isomorphic to either L or M.
To describe the above properties in the case where G is the direct sum of an infinite set of subgroups, more care is needed. If g is an element of the cartesian product ∏ of a set of groups, let gi be the ith element of g in the product. The external direct sum of a set of groups is the subset of ∏, where, for each element g of ∑E, gi is the identity for all but a finite number of gi. 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 the external direct sum is equal to the direct product. If G = ∑Hi, then G is isomorphic to ∑E. 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 a unique set such that g = ∏.
