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 {H_i} if

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 non-trivial subgroups is called decomposable; otherwise 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 = ∑H_i, where {H_i} is a finite set of subgroups.

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}.

...
Wikipedia