# Direct sum of groups

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In mathematics, a group *G* is called the **direct sum** ^{[1]}^{[2]} 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 in common), and *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*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*=*h*_{1}**h*_{2}* ... **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*= (*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, ∑*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

- If is a divisible subgroup of an abelian group . Then there exist another subgroup such that

- I is also a vector space then can be writen 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, *V*_{4} = *C*_{2} × *C*_{2}, we have that

*V*_{4}= <(0,1)> + <(1,0)> and*V*_{4}= <(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 *i*th 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*}.