Direct sum of groups

In group theory, a group G is called the direct sum of a set of subgroups {H_i} if

• each H_i is a normal subgroup of G
• each distinct pair of subgroups has trivial intersection, 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 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 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:

• 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 = h₁*h₂* ... * 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 = (h₁,h₂, ..., 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}.

Equivalence of direct sums

The direct sum is not unique for a group; for example, in the Klein group, V₄ = C₂ × C₂, we have that

V₄ = <(0,1)> + <(1,0)> and

V₄ = <(1,1)> + <(1,0)>.

However, it is the content of the Remak-Krull-Schmidt theorem that given a finite group G = ∑A_i = ∑B_j, where each A_i and each B_j is non-trivial and indecomposable, then the two sums are equivalent up to reordering and isomorphism of the subgroups involved.

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.

Generalization to sums over infinite sets

If we wish to describe the above properties in the case where G is the direct sum of an infinite (perhaps uncountable) set of subgroups, we need to be a bit more careful.

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}.