title: Direct sum of groups
text: # Direct sum of groups
  
  Algebraic structure → Group theory Group theory :
  
  Basic notions
  
  Subgroup
  
  Normal subgroup
  
  Quotient group
  
  (Semi-)direct product
  
  Group homomorphisms
  
  kernel
  
  image
  
  direct sum
  
  wreath product
  
  simple
  
  finite
  
  infinite
  
  continuous
  
  multiplicative
  
  additive
  
  cyclic
  
  abelian
  
  dihedral
  
  nilpotent
  
  solvable
  
  action
  
  Glossary of group theory
  
  List of group theory topics :
  
  Finite groups
  
  Classification of finite simple groups
  
  cyclic
  
  alternating
  
  Lie type
  
  sporadic
  
  Cauchy's theorem
  
  Lagrange's theorem
  
  Sylow theorems
  
  Hall's theorem
  
  p-group
  
  Elementary abelian group
  
  Frobenius group
  
  Schur multiplier
  
  Symmetric group $S_n$
  
  Klein four-group V
  
  Dihedral group $D_n$
  
  Quaternion group Q
  
  Dicyclic group $Dic_n$ :
  
  Discrete groupsLattices
  
  Integers ($\mathbb {Z}$)
  
  Free group
  
  Modular groups PSL(2, $\mathbb {Z}$)SL(2, $\mathbb {Z}$)
  
  Arithmetic group
  
  Lattice
  
  Hyperbolic group :
  
  Topological and Lie groups
  
  Solenoid
  
  Circle
  
  General linear GL(n)
  
  Special linear SL(n)
  
  Orthogonal O(n)
  
  Euclidean E(n)
  
  Special orthogonal SO(n)
  
  Unitary U(n)
  
  Special unitary SU(n)
  
  Symplectic Sp(n)
  
  $G_2$
  
  $F_4$
  
  $E_6$
  
  $E_7$
  
  $E_8$
  
  Lorentz
  
  Poincaré
  
  Conformal
  
  Diffeomorphism
  
  Loop
  
  Infinite dimensional Lie group O(∞)SU(∞)Sp(∞) :
  
  Algebraic groups
  
  Linear algebraic group
  
  Reductive group
  
  Abelian variety
  
  Elliptic curve :
  
  vte :
  
  In mathematics, a group G is called the direct sum of two normal subgroups with trivial intersection if it is generated by the subgroups. In abstract algebra, this method of construction of groups can be generalized to direct sums of vector spaces, modules, and other structures; see the article direct sum of modules for more information. A group which can be expressed as a direct sum of non-trivial subgroups is called decomposable, and if a group cannot be expressed as such a direct sum then it is called indecomposable.
  
  ## Definition
  
  A group G is called the direct sum of two subgroups $H_1$ and $H_2$ if
  
  * each $H_1$ and $H_2$ are normal subgroups of G,
  * the subgroups $H_1$ and $H_2$ have trivial intersection (i.e., having only the identity element $e$ of G in common),
  * G = $⟨H_1,$ $H_2⟩;$ in other words, G is generated by the subgroups $H_1$ and $H_2.$
  
  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 ≠ i}⟩,
  * 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, and if G is the direct sum of a set of subgroups ${H_i}$ then we often write G = $ΣH_i.$ Loosely speaking, a direct sum is isomorphic to a weak direct product of subgroups.
  
  ## Properties
  
  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 = $ΣH_i,$ 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 exists a unique set of elements $h_i$ in $H_i$ such that
  
      g = $h_1$ ∗ $h_2$ ∗ ... ∗ $h_i$ ∗ ... ∗ $h_n$
  * 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 the direct product, where each g can be expressed uniquely as
  
      g = $(h_1,h_2,$ ..., $h_i,$ ..., $h_n).$
  
  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, $ΣH_i$ is isomorphic to the direct product $×{H_i}.$
  
  ## Direct summand
  
  Given a group $G$, we say that a subgroup $H$ is a direct summand of $G$ if there exists another subgroup $K$ of $G$ such that $G=H+K$.
  
  In abelian groups, if $H$ is a divisible subgroup of $G$, then $H$ is a direct summand of $G$.
  
  ## Examples
  
  * If we take $G=∏_{i∈I}H_i{\textstyle$ G=\prod _{i\in I}H_{i}} it is clear that $G$ is the direct product of the subgroups $H_{i0}×∏_{i≠i0}H_i{\textstyle$ H_{i_{0}}\times \prod _{i\not =i_{0}}H_{i}}.
  * If $H$ is a divisible subgroup of an abelian group $G$ then there exists another subgroup $K$ of $G$ such that $G=K+H$.
  * If $G$ also has a vector space structure then $G$ can be written as a direct sum of $\mathbb {R}$ and another subspace $K$ that will be isomorphic to the quotient $G/K$.
  
  ## 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 $V_{4}\cong C_{2}\times C_{2}$ we have that
  
      $V_{4}=\langle (0,1)\rangle +\langle (1,0)\rangle ,$ and
      $V_{4}=\langle (1,1)\rangle +\langle (1,0)\rangle .$
  
  However, the Remak-Krull-Schmidt theorem states that given a finite group G = $ΣA_i$ = $ΣB_j,$ 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 conclude 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 $Π{H_i}$ 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 $Σ_E{H_i})$ is the subset of $Π{H_i},$ where, for each element g of $Σ_E{H_i},$ $g_i$ is the identity $e_{H_{i}}$ 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 equal to the direct product.
  
  If G = $ΣH_i,$ then G is isomorphic to $Σ_E{H_i}.$ 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 a unique set ${h_i$ ∈ $H_i$ : i ∈ S} such that g = Π ${h_i$ : i in S}.
  
  