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In [[mathematics]], a [[group (mathematics)|group]] ''G'' is called the '''direct sum''' of two [[subgroup]]s ''H''<sub>''1''</sub> and ''H''<sub>''2''</sub> if
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* 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
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* each ''H''<sub>''i''</sub> is a [[normal subgroup]] of ''G''
 
* each ''H''<sub>''i''</sub> has trivial intersection with the subgroup  <{''H''<sub>''j''</sub> : ''j'' not equal to ''i''}>, and
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* ''G'' = <{''H''<sub>''i''</sub>}>; in other words, ''G'' is [[generating set of a group|generated]] by the subgroups {''H''<sub>''i''</sub>}.
 
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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.
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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.
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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''.
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A group which can be expressed as a direct sum of non-trivial subgroups is called ''decomposable''; otherwise it is called ''indecomposable''.
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If ''G'' = ''H'' + ''K'', then it can be proven that:
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* for all ''h'' in ''H'', ''k'' in ''K'', we have that ''h''*''k'' = ''k''*''h''
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* 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''
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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.
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* 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>
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* for each ''g'' in ''G'', there unique set of {''h''<sub>''i''</sub> in ''H''<sub>''i''</sub>} such that
 
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* 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>
 
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Note the similarity with the [[direct product of groups|direct product]], where each ''g'' can be expressed uniquely as
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:''g'' = (''h''<sub>1</sub>,''h''<sub>2</sub>, ..., ''h''<sub>''i''</sub>, ..., ''h''<sub>''n''</sub>)


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>}.
United Suppliers, based in Eldora, Wi, is opening offices in Ames, Iowa bringing with them Sixty jobs  30 retained in addition to 30 new  and a investment investment totaling $3 million. These are a wholesale organization properties of agricultural retailers who do small business throughout the Midwest. <br><br>Idea For college kids 20: Run A Vending Appliance  Running a vending machine industry is like sitting and generating. Of course, a good location could well be required to run the business. When avid movie goers, a good choice would be to buy classic flicks or frame autographed posters of their most favorite celebrity. In addition, if they are into art, you could potentially buy them sculptures and artwork for the special occasion you are celebrating. <br><br>I have experienced being in space, while at the cinema previous to. When I worked at the Scientific discipline Museum I was able to put on their IMAX whenever they were showing Walking on the Moon: Animations. Hell I have been programming intended for 8+ years and even when I last but not least got into the business world I am Even now learning a fair bit. Cornerstone is everything, and you're not really going to lay a solid a single for software development in your years time unless that individual is a damn prodigy in the first place.. <br><br>Tips for high altitude cooking throughout Breckenridge can be found in several specialty publications. Some easy tips to adhere to are to use additional waters or other cooking such as butter or oil, and pump up the high temperature, maybe 20 30 stages. Third [http://www.gloucestercottage.co.nz/catalogue/parse.asp?u=70-Ugg-Boots-Nz-Wellington Ugg Boots Nz Wellington] and finally, I accomplished some work on the insurance quotes section. Under > Quotes you at this moment find a cascade of subcategories. <br><br>A recent favorite of Isle would be the Lot 200 Cameron Hughes 2007 Napa Valley Cabernet Sauvignon, which costs $28 a bottle at Costco. Simply just tasted this two days ago, and it also was pretty damn good cabernet for the price, Tropical isle says.. Btw. Intel provides sandy+ivy bridge for the same price normally  there are just too many sandy fill cpus out there. <br><br>Eventually, a love of learning and a few thoughtful teachers helped him turn away from the lure [http://www.thebluepub.co.nz/templates/Express/green/enquiries.asp?k=130-Nike-New-Zealand Nike New Zealand] of drugs and gang violence that permeated his neighborhood. Many of their friends weren't so lucky, ending up dead or in dejecting prison. Allen et Us middot Treatment Center Frequently asked questions. This isn 39 Really Work Scholar Employment. <br><br>Often known as the goddess of beauty and abundance, Lakshmi can often be evoked for matters of money, enjoy and good health. Because of the girl's connection with more worldly desires, I found myself reluctant to help [http://www.avantek.com.sg/images/slideshow/circle.asp?ad=49-Nike-Free-Run-3.0-V4 Nike Free Run 3.0 V4] Lakshmi. This is one of the biggest mistakes I've ever made concerning a songs purchase. It wasn until many years later that I heard about this kind [http://www.milfordflights.co.nz/includes/flights.asp?r=40-Cheap-Nike-Roshe-Run-Shoes Cheap Nike Roshe Run Shoes] of strange band called Push By Truckers again obtained read a few reviews in a very couple magazines of their cd Rock Opera which was extremely spoken of.<ul>
 
 
==Equivalence of decompositions into direct sums==
  <li>[http://leyyou.cn/forum.php?mod=viewthread&tid=170374 http://leyyou.cn/forum.php?mod=viewthread&tid=170374]</li>
 
 
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
  <li>[http://www.dailymenanews.com/node/37069 http://www.dailymenanews.com/node/37069]</li>
:''V''<sub>4</sub> = <(0,1)> + <(1,0)> and
 
:''V''<sub>4</sub> = <(1,1)> + <(1,0)>.
  <li>[http://www.57162.com/forum.php?mod=viewthread&tid=266962&fromuid=56173 http://www.57162.com/forum.php?mod=viewthread&tid=266962&fromuid=56173]</li>
 
 
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.
  <li>[http://www.seabear.com.hk/forum/read.php?tid=18675 http://www.seabear.com.hk/forum/read.php?tid=18675]</li>
 
 
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''.
  <li>[http://www.my9ye.com/news/html/?32667.html http://www.my9ye.com/news/html/?32667.html]</li>
 
 
==Generalization to sums over infinite sets==
</ul>
 
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]]
 
{{DEFAULTSORT:Direct Sum Of Groups}}
[[Category:Group theory]]

Revision as of 16:35, 7 February 2014

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