Direct sum of groups

In mathematics, a group G is called the direct sum of two subgroups H1 and H2 if More generally, G is called the direct sum of a finite set of subgroups {Hi} if 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 {Hi} 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 commutative up 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 + (L + M) = 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: The above assertions can be generalized to the case of G = ∑Hi, where {Hi} is a finite set of subgroups: