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{{Cleanup-rewrite|several issues are raised on the discussion page|date=March 2013}}
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.


{{Group theory sidebar |Basics}}
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]
* Only registered users will be able to execute this rendering mode.
* Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.


In [[mathematics]], a [[group (mathematics)|group]] ''G'' is called the '''direct sum''' <ref>Homology. Saunders MacLane. Springer, Berlin; Academic Press, New York, 1963.</ref><ref>László Fuchs. Infinite Abelian Groups</ref> of two [[subgroup]]s ''H''<sub>''1''</sub> and ''H''<sub>''2''</sub> if
Registered users will be able to choose between the following three rendering modes:
* each ''H''<sub>''1''</sub> and ''H''<sub>''2''</sub> are [[normal subgroup]]s of ''G''
* the subgroups ''H''<sub>''1''</sub> and ''H''<sub>''2''</sub> have [[Trivial group|trivial intersection]] (i.e., having only the [[identity element]] <math>e</math> in common), and
* ''G'' = <''H''<sub>''1''</sub>, ''H''<sub>''2''</sub>>; in other words, ''G'' is [[generating set of a group|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
'''MathML'''
* each ''H''<sub>''i''</sub> is a [[normal subgroup]] of ''G''
:<math forcemathmode="mathml">E=mc^2</math>
* each ''H''<sub>''i''</sub> has trivial intersection with the subgroup  <{''H''<sub>''j''</sub> : ''j'' not equal to ''i''}>, and
* ''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 ''G'' = ''H'' + ''K''; if ''G'' is the direct sum of a set of subgroups {''H''<sub>''i''</sub>}, we often write ''G'' = ∑''H''<sub>''i''</sub>. Loosely speaking, a direct sum is [[isomorphism|isomorphic]] to a weak direct product of subgroups.
<!--'''PNG''' (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


In [[abstract algebra]], this method of construction 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.
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


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''.
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].


A group which can be expressed as a direct sum of non-trivial subgroups is called ''decomposable''; otherwise it is called ''indecomposable''.
==Demos==


If ''G'' = ''H'' + ''K'', then it can be proven that:
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


* 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''<sub>''i''</sub>, where {''H''<sub>i</sub>} is a finite set of subgroups.
* accessibility:
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


* if ''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 ''h''<sub>''i''</sub> * ''h''<sub>''j''</sub> = ''h''<sub>''j''</sub> * ''h''<sub>''i''</sub>
==Test pages ==
* for each ''g'' in ''G'', there unique set of {''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 ((∑''H''<sub>''i''</sub>) + ''K'')/''K'' is isomorphic to ∑''H''<sub>''i''</sub>


Note the similarity with the [[direct product of groups|direct product]], where each ''g'' can be expressed uniquely as
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
:''g'' = (''h''<sub>1</sub>,''h''<sub>2</sub>, ..., ''h''<sub>''i''</sub>, ..., ''h''<sub>''n''</sub>)
*[[Displaystyle]]
*[[MathAxisAlignment]]
*[[Styling]]
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]


Since ''h''<sub>''i''</sub> * ''h''<sub>''j''</sub> = ''h''<sub>''j''</sub> * ''h''<sub>''i''</sub> 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''<sub>''i''</sub> is isomorphic to the direct product &times;{''H''<sub>''i''</sub>}.
*[[Inputtypes|Inputtypes (private Wikis only)]]
 
*[[Url2Image|Url2Image (private Wikis only)]]
==Direct summand==
==Bug reporting==
Given a group <math>G</math>, we say that a subgroup <math>H</math> is a '''direct summand''' of <math>G</math> (or that '''splits''' form <math>G</math>) if and only if there exist another subgroup <math>K\leq G</math> such that <math>G</math> is the direct sum of the subgroups <math>H</math>  and <math>K</math>
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
 
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> G= \prod_{i\in I} H_i </math> it is clear that <math> G </math> is the direct product of the subgroups <math> 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 exist another subgroup <math>K\leq G </math> such that  <math>G=K+H </math>
 
*I <math>G</math> is also a [[vector space]] then <math>G</math> can be writen as a direct sum of <math>\mathbb R</math> and another subespace <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]], ''V''<sub>4</sub> = ''C''<sub>2</sub> &times; ''C''<sub>2</sub>, we have that
:''V''<sub>4</sub> = <(0,1)> + <(1,0)> and
:''V''<sub>4</sub> = <(1,1)> + <(1,0)>.
 
However, it is the content of the [[Remak-Krull-Schmidt theorem]] that given a finite group ''G'' = ∑''A''<sub>''i''</sub> = ∑''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 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''<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 ∑<sub>'''''E'''''</sub>{''H''<sub>''i''</sub>}) is the subset of ∏{''H''<sub>''i''</sub>}, where, for each element ''g'' of ∑<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 identical to the direct product.
 
If ''G'' = ∑''H''<sub>''i''</sub>, then ''G'' is isomorphic to ∑<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 unique {''h''<sub>''i''</sub> in ''H''<sub>''i''</sub> : ''i'' in ''S''} such that ''g'' = ∏ {''h''<sub>''i''</sub> : ''i'' in ''S''}.
 
==See also==
*[[direct sum]]
*[[coproduct]]
*[[free product]]
*[[Direct sum of topological groups]]
 
==References==
{{Reflist}}
 
{{DEFAULTSORT:Direct Sum Of Groups}}
[[Category:Group theory]]

Latest revision as of 22:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML

E=mc2


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

If you find any bugs, please report them at Bugzilla, or write an email to math_bugs (at) ckurs (dot) de .

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