Direct sum of groups

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In mathematics, a group G is called the direct sum ^[1][2] of two subgroups H₁ and H₂ if

• each H₁ and H₂ are normal subgroups of G
• the subgroups H₁ and H₂ have trivial intersection (i.e., having only the identity element in common), and
• G = <H₁, H₂>; in other words, G is generated by the subgroups H₁ and H₂.

More generally, G is called the direct sum of a finite set of subgroups {H_i} if

• each H_i is a normal subgroup of G
• each H_i has trivial intersection with the subgroup <{H_j : j not equal to i}>, 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 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:

• 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}.

Direct summand

Given a group , we say that a subgroup is a direct summand of (or that splits form ) if and only if there exist another subgroup such that is the direct sum of the subgroups and

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 exist another subgroup such that
• If is also a vector space then can be written as a direct sum of and another subespace 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, 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, 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 then assume that H is isomorphic to either L or M.

Generalization to sums over infinite sets

To describe the above properties in the case where G is the direct sum of an infinite (perhaps uncountable) set of subgroups, more care is needed.

If g is an element of the cartesian product ∏{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 ∏{H_i}, where, for each element g of ∑_E{H_i}, g_i is the identity 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.

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.

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. 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 = ∏ {h_i : i in S}.

See also

• direct sum
• coproduct
• free product
• Direct sum of topological groups

References