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Algebraic structure →Group theory Group theory
Basic notions Subgroup Normal subgroup Group action Quotient group (Semi-)direct product Direct sum Free product Wreath product Group homomorphisms kernel image simple finite infinite continuous multiplicative additive cyclic abelian dihedral nilpotent solvable Glossary of group theory List of group theory topics
Finite groups Cyclic group Zn Symmetric group Sn Alternating group An Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group Schur multiplier Classification of finite simple groups cyclic alternating Lie type sporadic
Discrete groups Lattices Integers (${\displaystyle \mathbb{Z}}$) Free group Modular groups PSL(2,${\displaystyle \mathbb{Z}}$) SL(2,${\displaystyle \mathbb{Z}}$) Arithmetic group Lattice Hyperbolic group
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Inmathematics, agroupG is called thedirect sum^[1][2] of twonormal subgroups withtrivial intersection if it isgenerated by the subgroups. Inabstract algebra, this method of construction of groups can be generalized to direct sums ofvector spaces,modules, and other structures; see the articledirect sum of modules for more information. A group which can be expressed as a direct sum of non-trivial subgroups is calleddecomposable, and if a group cannot be expressed as such a direct sum then it is calledindecomposable.

AgroupG is called thedirect sum^[1][2] of twosubgroupsH₁ andH₂ if

• eachH₁ andH₂ are normal subgroups ofG,
• the subgroupsH₁ andH₂ have trivial intersection (i.e., having only theidentity element${\displaystyle e}$ ofG in common),
• G = ⟨H₁,H₂⟩; in other words,G is generated by the subgroupsH₁ andH₂.

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

• eachH_i is anormal subgroup ofG,
• eachH_i has trivial intersection with the subgroup⟨{H_j :j ≠i}⟩,
• G = ⟨{H_i}⟩; in other words,G isgenerated by the subgroups {H_i}.

IfG is the direct sum of subgroupsH andK then we writeG =H +K, and ifG is the direct sum of a set of subgroups {H_i} then we often writeG = ΣH_i. Loosely speaking, a direct sum isisomorphic to a weak direct product of subgroups.

IfG =H +K, then it can be proven that:

• for allh inH,k inK, we have thath ∗k =k ∗h
• for allg inG, there exists uniqueh inH,k inK such thatg =h ∗k
• There is a cancellation of the sum in a quotient; so that(H +K)/K is isomorphic toH

The above assertions can be generalized to the case ofG = ΣH_i, where {H_i} is a finite set of subgroups:

• ifi ≠j, then for allh_i inH_i,h_j inH_j, we have thath_i ∗h_j =h_j ∗h_i
• for eachg inG, there exists a unique set of elementsh_i inH_i such that

g =h₁ ∗h₂ ∗ ... ∗h_i ∗ ... ∗h_n

• There is a cancellation of the sum in a quotient; so that((ΣH_i) +K)/K is isomorphic to ΣH_i.

Note the similarity with thedirect product, where eachg can be expressed uniquely as

g = (h₁,h₂, ...,h_i, ...,h_n).

Sinceh_i ∗h_j =h_j ∗h_i for alli ≠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, ΣH_i is isomorphic to the direct product ×{H_i}.

Given a group${\displaystyle G}$, we say that a subgroup${\displaystyle H}$ is adirect summand of${\displaystyle G}$ if there exists another subgroup${\displaystyle K}$ of${\displaystyle G}$ such that${\displaystyle G=H+K}$.

In abelian groups, if${\displaystyle H}$ is adivisible subgroup of${\displaystyle G}$, then${\displaystyle H}$ is a direct summand of${\displaystyle G}$.

• If we take$G=\prod _{i\in I}{H}_i$ it is clear that${\displaystyle G}$ is the direct product of the subgroups$H_{i_0}\times \prod _{i\ne i_0}{H}_i$.

• If${\displaystyle H}$ is adivisible subgroup of an abelian group${\displaystyle G}$ then there exists another subgroup${\displaystyle K}$ of${\displaystyle G}$ such that${\displaystyle G=K+H}$.
• If${\displaystyle G}$ also has avector space structure then${\displaystyle G}$ can be written as a direct sum of${\displaystyle \mathbb{R}}$ and another subspace${\displaystyle K}$ that will be isomorphic to the quotient${\displaystyle G/K}$.

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 theKlein group${\displaystyle V_4\cong C_2\times C_2}$ we have that

${\displaystyle V_4=⟨(0,1)⟩+⟨(1,0)⟩,}$ and

${\displaystyle V_4=⟨(1,1)⟩+⟨(1,0)⟩.}$

However, theRemak-Krull-Schmidt theorem states that given afinite groupG = ΣA_i = ΣB_j, where eachA_i and eachB_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 infiniteG =H +K =L +M, even when all subgroups are non-trivial and indecomposable, we cannot conclude thatH is isomorphic to eitherL orM.

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

Ifg is an element of thecartesian product Π{H_i} of a set of groups, letg_i be theith element ofg in the product. Theexternal direct sum of a set of groups {H_i} (written as Σ_E{H_i}) is the subset of Π{H_i}, where, for each elementg of Σ_E{H_i},g_i is the identity${\displaystyle e_{H_i}}$ for all but a finite number ofg_i (equivalently, only a finite number ofg_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 equal to the direct product.

IfG = ΣH_i, thenG is isomorphic to Σ_E{H_i}. Thus, in a sense, the direct sum is an "internal" external direct sum. For each elementg inG, there is a unique finite setS and a unique set {h_i ∈H_i :i ∈S} such thatg = Π {h_i :i inS}.

• Direct sum
• Coproduct
• Free product
• Direct sum of topological groups

• Group theory

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