Editing Direct sum of groups

{{Use American English|date = January 2019}}
{{Short description|Means of constructing a group from two subgroups}}
{{Cleanup rewrite|several issues are raised on the discussion page|date=March 2013}}
{{Group theory sidebar |Basics}}

In [[mathematics]], a [[group (mathematics)|group]] ''G'' is called the '''direct sum'''<ref name=":0">Homology. Saunders MacLane. Springer, Berlin; Academic Press, New York, 1963.</ref><ref name=":1">László Fuchs. Infinite Abelian Groups</ref> of two [[Normal subgroup|normal subgroups]] with [[Trivial group|trivial intersection]] if it is [[Generating set of a group|generated]] by the subgroups. In [[abstract algebra]], this method of construction of groups can be generalized to direct sums of [[vector space]]s, [[module (mathematics)|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 (mathematics)|group]] ''G'' is called the '''direct sum'''<ref name=":0" /><ref name=":1" /> of two [[subgroup]]s ''H''<sub>1</sub> and ''H''<sub>2</sub> if
* each ''H''<sub>1</sub> and ''H''<sub>2</sub> are normal subgroups of ''G'',
* the subgroups ''H''<sub>1</sub> and ''H''<sub>2</sub> have trivial intersection (i.e., having only the [[identity element]] <math>e</math> of ''G'' in common),
* ''G'' = ⟨''H''<sub>1</sub>, ''H''<sub>2</sub>⟩; in other words, ''G'' is generated by the subgroups ''H''<sub>1</sub> and ''H''<sub>2</sub>.

More generally, ''G'' is called the  direct sum of a finite set of [[subgroup]]s {''H''<sub>''i''</sub>} if
* each ''H''<sub>''i''</sub> is a [[normal subgroup]] of ''G'',
* each ''H''<sub>''i''</sub> has trivial intersection with the subgroup {{nowrap|⟨{''H''<sub>''j''</sub> : ''j'' ≠ ''i''}⟩}},
* ''G'' = ⟨{''H''<sub>''i''</sub>}⟩; in other words, ''G'' is [[generating set of a group|generated]] by the subgroups {''H''<sub>''i''</sub>}.

If ''G'' is the direct sum of subgroups ''H'' and ''K'' then we write {{nowrap|1=''G'' = ''H'' + ''K''}}, and if ''G'' is the direct sum of a set of subgroups {''H''<sub>''i''</sub>} then we often write ''G'' = Σ''H''<sub>''i''</sub>. Loosely speaking, a direct sum is [[isomorphism|isomorphic]] to a weak direct product of subgroups.

== Properties ==
If {{nowrap|1=''G'' = ''H'' + ''K''}}, then it can be proven that:

* for all ''h'' in ''H'', ''k'' in ''K'', we have that {{nowrap|1=''h'' ∗ ''k'' = ''k'' ∗ ''h''}}
* for all ''g'' in ''G'', there exists unique ''h'' in ''H'', ''k'' in ''K'' such that {{nowrap|1=''g'' = ''h'' ∗ ''k''}}
* There is a cancellation of the sum in a quotient; so that {{nowrap|(''H'' + ''K'')/''K''}} is isomorphic to ''H''

The above assertions can be generalized to the case of {{nowrap|1=''G'' = Σ''H''<sub>''i''</sub>}}, where {''H''<sub>i</sub>} is a finite set of subgroups:

* if {{nowrap|''i'' ≠ ''j''}}, then for all ''h''<sub>''i''</sub> in ''H''<sub>''i''</sub>, ''h''<sub>''j''</sub> in ''H''<sub>''j''</sub>, we have that {{nowrap|1=''h''<sub>''i''</sub> ∗ ''h''<sub>''j''</sub> = ''h''<sub>''j''</sub> ∗ ''h''<sub>''i''</sub>}}
* for each ''g'' in ''G'', there exists a unique set of elements ''h''<sub>''i''</sub> in ''H''<sub>''i''</sub> such that
:''g'' = ''h''<sub>1</sub> ∗ ''h''<sub>2</sub> ∗ ... ∗ ''h''<sub>''i''</sub> ∗ ... ∗ ''h''<sub>''n''</sub>
* There is a cancellation of the sum in a quotient; so that {{nowrap|((Σ''H''<sub>''i''</sub>) + ''K'')/''K''}} is isomorphic to Σ''H''<sub>''i''</sub>.

Note the similarity with the [[direct product of groups|direct product]], where each ''g'' can be expressed uniquely as
:''g'' = (''h''<sub>1</sub>,''h''<sub>2</sub>, ..., ''h''<sub>''i''</sub>, ..., ''h''<sub>''n''</sub>).

Since {{nowrap|1=''h''<sub>''i''</sub> ∗ ''h''<sub>''j''</sub> = ''h''<sub>''j''</sub> ∗ ''h''<sub>''i''</sub>}} for all {{nowrap|''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, Σ''H''<sub>''i''</sub> is isomorphic to the direct product ×{''H''<sub>''i''</sub>}.

==Direct summand==
Given a group <math>G</math>, we say that a subgroup <math>H</math> is a '''direct summand''' of <math>G</math> if there exists another subgroup <math>K</math> of <math>G</math> such that <math>G = H+K</math>.

In abelian groups, if <math>H</math> is a [[Divisible group|divisible subgroup]] of <math>G</math>, then <math>H</math> is a direct summand of <math>G</math>.

==Examples==

* If we take  <math display="inline"> G= \prod_{i\in I} H_i </math> it is clear that <math> G </math> is the direct product of the subgroups <math display="inline"> H_{i_0} \times \prod_{i\not=i_0}H_i</math>.

* If <math>H</math> is a [[Divisible group|divisible subgroup]] of an abelian group <math>G</math> then there exists another subgroup <math>K</math> of <math>G</math> such that <math>G=K+H</math>.
* If <math>G</math> also has a [[vector space]] structure then <math>G</math> can be written as a direct sum of <math>\mathbb R</math> and another subspace <math>K</math> that will be isomorphic to the quotient <math>G/K</math>.

==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]] <math>V_4 \cong C_2 \times C_2</math> we have that
: <math>V_4 = \langle(0,1)\rangle + \langle(1,0)\rangle,</math> and
: <math>V_4 = \langle(1,1)\rangle + \langle(1,0)\rangle.</math>

However, the [[Remak-Krull-Schmidt theorem]] states that given a ''finite'' group ''G'' = Σ''A''<sub>''i''</sub> = Σ''B''<sub>''j''</sub>, where each ''A''<sub>''i''</sub> and each ''B''<sub>''j''</sub> 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''<sub>''i''</sub>} of a set of groups, let ''g''<sub>''i''</sub> be the ''i''th element of ''g'' in the product. The '''external direct sum''' of a set of groups {''H''<sub>''i''</sub>} (written as Σ<sub>'''''E'''''</sub>{''H''<sub>''i''</sub>}) is the subset of Π{''H''<sub>''i''</sub>}, where, for each element ''g'' of Σ<sub>'''''E'''''</sub>{''H''<sub>''i''</sub>}, ''g''<sub>''i''</sub> is the identity <math>e_{H_i}</math> for all but a finite number of ''g''<sub>''i''</sub> (equivalently, only a finite number of ''g''<sub>''i''</sub> 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''<sub>''i''</sub>} the external direct sum is equal to the direct product.

If ''G'' = Σ''H''<sub>''i''</sub>, then ''G'' is isomorphic to Σ<sub>'''''E'''''</sub>{''H''<sub>''i''</sub>}. 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''<sub>''i''</sub> ∈ ''H''<sub>''i''</sub> : ''i'' ∈ ''S''} such that ''g'' = Π {''h''<sub>''i''</sub> : ''i'' in ''S''}.

==See also==
*[[Direct sum]]
*[[Coproduct]]
*[[Free product]]
*[[Direct sum of topological groups]]

==References==
{{Reflist}}

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[[Category:Group theory]]