In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a group

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

 G is called the direct sum of a set of subgroup

Subgroup

In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...

s {H_i} if

• each H_i is a normal subgroup
  Normal subgroup
  In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....
  
   of G
• each distinct pair of subgroups has trivial intersection, and
• G = <{H_i}>; in other words, G is generated
  Generating set of a group
  In abstract algebra, a generating set of a group is a subset that is not contained in any proper subgroup of the group. Equivalently, a generating set of a group is a subset such that every element of the group can be expressed as the combination of finitely many elements of the subset and their...
  
   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

Isomorphism

In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations.  If there exists an isomorphism between two structures, the two structures are said to be isomorphic.  In a certain sense, isomorphic structures are...

 to a weak direct product of subgroups.

In abstract algebra

Abstract algebra

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...

, this method of construction can be generalized to direct sums of vector space

Vector space

A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

s, modules

Module (mathematics)

In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...

, and other structures; see the article direct sum of modules

Direct sum of modules

In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The result of the direct summation of modules is the "smallest general" module which contains the given modules as submodules...

 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

Direct product of groups

In the mathematical field of group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted...

, 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

Cartesian product

In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...

∏{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.

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 = ∏ {h_i : i in S}.