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 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 non-trivial subgroups is called decomposable; otherwise it is called indecomposable.
If G = H + K, then it can be proven that:
Table of contents |
2 Generalization to sums over infinite sets 3 Subdirect products and subdirect 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
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 then assume that H is isomorphic to either L or M.
If g is an element of the cartesian product π{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 π{H_{i}}, where, for each element g of ∑_{E}{H_{i}}), g_{i} is the identity e_{Hi} 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.
It should be readily apparent that 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.
Then 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. We have that, 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 = π {h_{i} : i in S}