Exclusive or: Difference between revisions

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When it comes to standing out of crowd the - Nicki Minaj knows how to just achieve it. "The Wolverine" is a vast improvement over "X-Men Origins: Wolverine. This doesn't necessarily mean eating up your greens like your mother told you. No: from that moment I declared everlasting war against the species, and, more than all, against him who had formed me and sent me forth to this insupportable misery. The end result is that the male who is suffering these problems will be able to lead a happy and sexually satisfying life. <br><br>Today, thanks to new surgical techniques that spare the nerves and vessels feeding the penis, erectile dysfunction is far from inevitable. Who think that even though we're writing about the internet, we should still make an effort to be entertaining and erudite. "However, anywhere there exists a body art, since your grandmother may have mentioned, trouble's never significantly powering. You should not take Cialis if you are taking a nitrate drug for chest pain or heart problems, including nitroglycerin. Guys who have achieved this are invaluable to research because they have served as the missing control group in studies; those without the effects of internet pornography: Previously, researchers couldn't find ANY guys who were not surfing pornography. <br><br>Fortunately, many patients regain potency 6 to 12 months after treatment. It acts by increasing the level of the gas nitric oxide in the blood, which helps certain muscles to relax - that in turn improves the blood circulation and the end effect is a strong and sustainable erection. They were also five times as likely to experience a live birth. After trying it once, you may find your partner requesting you keep it on. It hastens the growth of hair and stops the appearance of split ends. <br><br>Enzymatic tests have shown Red Kwao Krua to be a potent inhibitor of c-AMP Phosphodiesterase which reacts directly at the corpus cavernosum in the penis enhancing blood flow to that area creating a more frequent, longer lasting, stronger male sexual arousal period. The high selling price of the horns has brought about a sophisticated breed of poachers. Hormonal Imbalance: having low levels of testosterone contributes to impotence 8. Often cause for pity, this problem can usually be handled if the actual problem is discovered. Drinking plenty of water is also extremely beneficial. <br><br>If the symptoms are more serious and persistent, oral medication is prescribed. Often feed walnut porridge, can not only, still can prevent kidney impotence, spermatorrhea, have delayed failure anti-ageing effect. Not all studies point to such benefits, however, and many conventional doctors remain skeptical. Therefore, cheap generic Viagra have become famous. Follow the instructions on container for the required dosage.<br><br>If you liked this post and you would such as to receive additional info concerning [http://www.eiaculazione-precoce.info/ cures for impotence] kindly see the website.
{{unreferenced|date=December 2007}}
 
{{Group theory sidebar |Basics}}
 
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
* 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
* 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
* ''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.
 
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.
 
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''<sub>''i''</sub>, where {''H''<sub>i</sub>} is a finite set of subgroups.
 
* 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>
* 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
:''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>}.
 
==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]]
 
{{DEFAULTSORT:Direct Sum Of Groups}}
[[Category:Group theory]]

Latest revision as of 06:35, 15 December 2014

When it comes to standing out of crowd the - Nicki Minaj knows how to just achieve it. "The Wolverine" is a vast improvement over "X-Men Origins: Wolverine. This doesn't necessarily mean eating up your greens like your mother told you. No: from that moment I declared everlasting war against the species, and, more than all, against him who had formed me and sent me forth to this insupportable misery. The end result is that the male who is suffering these problems will be able to lead a happy and sexually satisfying life.

Today, thanks to new surgical techniques that spare the nerves and vessels feeding the penis, erectile dysfunction is far from inevitable. Who think that even though we're writing about the internet, we should still make an effort to be entertaining and erudite. "However, anywhere there exists a body art, since your grandmother may have mentioned, trouble's never significantly powering. You should not take Cialis if you are taking a nitrate drug for chest pain or heart problems, including nitroglycerin. Guys who have achieved this are invaluable to research because they have served as the missing control group in studies; those without the effects of internet pornography: Previously, researchers couldn't find ANY guys who were not surfing pornography.

Fortunately, many patients regain potency 6 to 12 months after treatment. It acts by increasing the level of the gas nitric oxide in the blood, which helps certain muscles to relax - that in turn improves the blood circulation and the end effect is a strong and sustainable erection. They were also five times as likely to experience a live birth. After trying it once, you may find your partner requesting you keep it on. It hastens the growth of hair and stops the appearance of split ends.

Enzymatic tests have shown Red Kwao Krua to be a potent inhibitor of c-AMP Phosphodiesterase which reacts directly at the corpus cavernosum in the penis enhancing blood flow to that area creating a more frequent, longer lasting, stronger male sexual arousal period. The high selling price of the horns has brought about a sophisticated breed of poachers. Hormonal Imbalance: having low levels of testosterone contributes to impotence 8. Often cause for pity, this problem can usually be handled if the actual problem is discovered. Drinking plenty of water is also extremely beneficial.

If the symptoms are more serious and persistent, oral medication is prescribed. Often feed walnut porridge, can not only, still can prevent kidney impotence, spermatorrhea, have delayed failure anti-ageing effect. Not all studies point to such benefits, however, and many conventional doctors remain skeptical. Therefore, cheap generic Viagra have become famous. Follow the instructions on container for the required dosage.

If you liked this post and you would such as to receive additional info concerning cures for impotence kindly see the website.

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