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{{about|the mathematical concept|the four-person anti-Nazi Resistance groups|Vierergruppe (German Resistance)}}
He got back late, and looked so tired I said I�d order a Rasa curry, which I did. So, on Friday, I emailed him in the morning to say that I�d been worried by the fact that he�d read the address of my London flat on the internet. They wanted to phone us back, so I reminded David I�d lost my BlackBerry, and have no idea what the number of the Bat Phone is.<br><br>I keep conjuring up images of him in 1983, trying to reignite the passion. <br>I told him, before he started wriggling, that I think that memorable evening, when after our game of squash he had asked me to take his racquet home for him because he had a date, he had already started seeing the woman he would marry.<br><br>He bought me a bottle of prosecco, and some shopping. On Sunday, we went to the Matisse exhibition at the Tate Modern (walking round the exhibit, talking, made me feel as though we were in a Woody Allen movie), and again his car had a parking ticket on it when we returned to it. This time, though, he didn�t hand it to me, although it�s sitting, accusingly, on my desk.<br><br>The whole phone, in fact, is a gleaming object of desire but it lacks standout new features other than the cameras, so you�ll miss out on gizmos such as the Samsung S5�s fingerprint scanner, or LG G3�s frankly frightening Quad HD screen.<br><br>99                  &#9733;&#9733;&#9733;&#9733;&#9733;Most of us who can be considered vaguely literate felt a faint anger when the term �selfie� passed from geek-speak into common parlance, especially after this year�s famous examples at the Oscars and Nelson Mandela�s funeral, where Barack Obama snuggled up to David Cameron. Huawei Ascend P7 �329.<br><br>And don�t say, �Don�t give me a hard time� when it�s you giving me a hard time. I did nothing today other than work hard and order dinner. 'Have a great life together, just leave me out of it. �You know I have no interest in her. I didn�t realise you had taken my keys back. I hope that was just a fit of pique. �<br>This came back the next morning, when he�d arrived at work. I love you and no one else. I have to work now, but I�ll see you tonight, as usual. My life is an open book to you.<br><br>Isobel and Dawn are in situ already, chilling the wine. Lots of books on Kindle. My Accessorize pink bikini. Wow, are we going to whip up some copy! What about the wedding proposal on the Pampelonne beach and me and Dawn can scatter white rose petals. Xxx� <br>The thing is, I�m not even sure David is still coming� Packing in tissue paper tonight: The Row sunglasses. My Dries negligee dress. Isobel has just sent me a message�<br>�The cast of Liz Jones�s Diary are off to the South of France. Let�s get this show on the road!<br><br>She had written to him three times, about him giving her his car (His reply: �I will send you the log book�), and having found his bow tie (His reply: �I spent �75 on one last week.<br><br>The famous Oscars �selfie� taken by Bradley Cooper and featuring Angelina Jolie, Brad Pitt, Meryl Streep, Julia Roberts, Ellen DeGeneres, Jennifer Lawrence, Lupita Nyong�o, her brother Peter, Kevin Spacey, Jared Leto and Channing Tatum<br>But Huawei (pronounced like the reverse of a jubilant �Whahey�) needed to add to the language to sum up the purpose of its new Ascend P7�s stand-out feature - a forward-facing eight-megapixel camera, with the option for panoramic shots. By law, this is the only phone you�ll be taking �groufies� on - although as yet, the trademark doesn�t apply in the UK, so users of other phones can still use it for their own work. Unless you�re the size of a Weight Watchers �before� picture, there�s only one reason for this to exist - a �group selfie� (ie, a group shot where one of you holds the camera) - hence �groufie�. Huawei is so proud of the word the company trademarked it in several countries to mark the launch of the P7.<br><br>In case you�re wondering what Huawei is, it�s one of those Chinese companies that only recently began hawking smartphones in the West, and shifts so many phones in the Far East it�s the third biggest phone company on Earth.<br><br>Upstage selfie-toting friends by turning you and your pals into a real 3D-model (warning: there�s a fair bit of work involved), ready to print off. The app �walks� you round anything to capture it in 3D - now all you need is a few hundred quid for a 3[https://www.Vocabulary.com/dictionary/D+printer D printer].<br><br>Huawei�s invention of the g-word, and the panoramic software to make it a reality, is down to a feeling that the endless Twitter parade of selfies (both celebrity and human), might be improved with a bit of context. And in action, it�s impressive too.<br><br>WOLFENSTEIN: THE NEW ORDER�40, PC, CONSOLES<br>The biggest surprise in Wolfensteing: The New Order is that it's the tense plotting that lifts this violent tale above its beige rivals <br>With an alternate-history plot hewn from the finest codswallop - a Nazi general uses high technology to summon an army of robots and zombies - the biggest surprise here is that it�s the tense plotting that lifts this violent tale above its beige rivals. &#9733;&#9733;&#9733;&#9733;&#9733;<br><br>If you adored this article and you would like to receive more info pertaining to [http://nouveauclashofclanstriche.blogspot.com/ http://nouveauclashofclanstriche.blogspot.com/] i implore you to visit our website.
{{Group theory sidebar}}
 
In [[mathematics]], the '''Klein four-group''' (or just '''Klein group''' or '''Vierergruppe''' ({{lang-en|four-group}}), often symbolized by the letter '''V''') is the [[Group (mathematics)|group]] Z<sub>2</sub> × Z<sub>2</sub>, the [[Direct product of groups|direct product]] of two copies of the [[cyclic group]] of [[Order (group theory)|order]] 2. It was named ''Vierergruppe'' by [[Felix Klein]] in his ''Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade'' in 1884.
 
The Klein four-group is the smallest non-[[cyclic group]]. It is given by the [[presentation of a group|group presentation]]
 
:<math>\mathrm{V}_4 = \langle a,b \mid a^2 = b^2 = (ab)^2 = 1 \rangle.</math>
 
All non-[[identity element|identity]] elements of the Klein group have order 2. It is [[abelian group|abelian]], and isomorphic to the [[dihedral group]] of order (cardinality) 4. It is also isomorphic to the [[direct sum]] {{nowrap|Z<sub>2</sub> ⊕ Z<sub>2</sub>}}, so that it can be represented as the bit strings {{nowrap|{00, 01, 10, 11} }} under [[bitwise XOR]].
 
The Klein group's [[Cayley table]] is given by:
 
::{| border="1" cellpadding="11" style="border-collapse: collapse; border: 1px #aaa solid;"
!style="background:#efefef;"| *
!style="background:#efefef;"| <big>1
!style="background:#efefef;"| <big>''a''
!style="background:#efefef;"| <big>''b''
!style="background:#efefef;"| <big>''ab''
|-
!style="background:#efefef;"| <big>1
| <big>1 || <big>''a'' || <big>''b'' || <big>''ab''
|-
!style="background:#efefef;"| <big>''a''
| <big>''a'' || <big>1 || <big>''ab'' || <big>''b''
|-
!style="background:#efefef;"| <big>''b''
| <big>''b'' || <big>''ab'' || <big>1 || <big>''a''
|-
!style="background:#efefef;"| <big>''ab''
| <big>''ab'' || <big>''b'' || <big>''a'' || <big>1
|}
 
An elementary construction of the Klein four-group is the [[multiplicative group]] {{nowrap|{ 1, 3, 5, 7 } }} with the action being multiplication [[modular arithmetic|modulo 8]]. Here ''a'' is 3, ''b'' is 5, and ''ab'' is {{nowrap|1=3 × 5 = 15 ≡ 7 (mod 8)}}.
 
In 2D it is the [[symmetry group]] of a [[rhombus]] and of a [[rectangle]] which are not [[square (geometry)|squares]], the four elements being the identity, the vertical reflection, the horizontal reflection, and a 180 degree rotation.
[[File:GreenRectangularCross.png|thumb|right|The symmetry group of this cross is the Klein four-group. It can be flipped horizontally or vertically and remain unchanged. Unlike a square, though, a quarter-turn rotation will change the figure.]]
 
In 3D there are three different symmetry groups that are algebraically the Klein four-group V:
*one with three perpendicular 2-fold rotation axes: D<sub>2</sub>
*one with a 2-fold rotation axis, and a perpendicular plane of reflection: {{nowrap|1=C<sub>2''h''</sub> = D<sub>1''d''</sub>}}
*one with a 2-fold rotation axis in a plane of reflection (and hence also in a perpendicular plane of reflection): {{nowrap|1=C<sub>2''v''</sub> = D<sub>1''h''</sub>}}.
 
The three elements of order 2 in the Klein four-group are interchangeable: the [[automorphism group]] is the group of permutations of the three elements. This  essential symmetry can also be seen by its [[Permutation representation (symmetric group)|permutation representation]] on 4 points:
[[File:Klein four-group; Cayley table; subgroup of S4 (elements 0,7,16,23).svg|thumb|Identity and double-[[transposition (mathematics)|transpositions]] of four objects form '''V''']]
 
[[File:Klein four-group; Cayley table; subgroup of S4 (elements 0,1,6,7).svg|thumb|Other permutations of four objects, forming '''V''' as well<br><br>See: [[Subgroup#4 elements|4 element subsets of S<sub>4</sub>]]]]
 
:'''V''' = { (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) }
 
In this representation, V is a [[normal subgroup]] of the [[alternating group]] A<sub>4</sub>
(and also the [[symmetric group]] S<sub>4</sub>) on 4 letters. In fact, it is the kernel of a surjective map from S<sub>4</sub> to S<sub>3</sub>.
According to [[Galois theory]], the existence of the Klein four-group (and in particular, this representation of it) explains the existence of the formula for calculating the roots of [[quartic equation]]s in terms of [[Radical of an algebraic group|radical]]s, as established by [[Lodovico Ferrari]]:
the map {{nowrap|S<sub>4</sub> → S<sub>3</sub>}} corresponds to the resolvent cubic, in terms of [[Lagrange resolvents]].
 
The Klein four-group as a subgroup of A<sub>4</sub> is not the [[automorphism group]] of any [[Graph (mathematics)|simple graph]]. It is, however, the automorphism group of a two-vertex graph where the vertices are connected to each other with ''two'' edges, making the graph non-simple. It is also the automorphism group of the following simple graph, but in the permutation representation {{nowrap|{ (), (1,2), (3,4), (1,2)(3,4) } }} where the points are labeled top-left, bottom-left, top-right, bottom-right:
::[[Image:Klein 4-Group Graph.svg|50px]]
 
In the construction of [[finite ring]]s, eight of the eleven rings with four elements have the Klein four-group as their additive substructure.
 
In [[music composition]] the four-group is the basic group of permutations in the [[twelve-tone technique]]. In that instance the Cayley table is written;<ref>Babbitt, Milton. (1960) "Twelve-Tone Invariants as Compositional Determinants", ''Musical Quarterly'' 46(2):253 Special Issue: Problems of Modern Music: The Princeton Seminar in Advanced Musical Studies (April): 246–59, [[Oxford University Press]]</ref>
{| border=1
|S
|I:
|R:
|RI:
|-
|I:
|S
|RI
|R
|-
|R:
|RI
|S
|I
|-
|RI:
|R
|I
|S
|}
 
In [[interval analysis]] the set of intervals [''a'',''b''] on the [[real line]] is identified with the points (''a'',''b'') of the [[Cartesian plane]] with [[Euclidean distance]].  The Klein four-group arises from the multiplicative structure on the direct sum {{nowrap|''R'' ⊕ ''R''}}. The x-axis and the y-axis are [[ideal (ring theory)|ideal]]s of the [[ring (mathematics)|ring]] of intervals where [1, 1] is the [[identity element|identity]] so [[quadrant (plane geometry)|quadrant]] I is the [[identity component|component of the identity]]. The Klein four-group is the [[quotient group|quotient]] of the multiplicative group of intervals that do not end on an axis by the subgroup consisting of intervals represented by points in quadrant I. Similarly, the ring of intervals is normalized to the [[split-complex number]] ring which also produces the Klein four-group as the quotient of its [[group of units]] by its identity component.
 
==See also==
*[[Quaternion group]]
 
==References==
{{Reflist}}
 
==Further reading==
* M. A. Armstrong (1988) ''Groups and Symmetry'', [[Springer Verlag]], {{Google books quote|id=lDfcBPl9Cu4C|pg=PA53|text=Klein's group|page 53}}.
* W. E. Barnes (1963) ''Introduction to Abstract Algebra'', D.C. Heath & Co., page 20.
* {{mathworld | urlname = Vierergruppe  | title = Vierergruppe }}
 
[[Category:Finite groups]]

Latest revision as of 16:14, 7 January 2015

He got back late, and looked so tired I said I�d order a Rasa curry, which I did. So, on Friday, I emailed him in the morning to say that I�d been worried by the fact that he�d read the address of my London flat on the internet. They wanted to phone us back, so I reminded David I�d lost my BlackBerry, and have no idea what the number of the Bat Phone is.

I keep conjuring up images of him in 1983, trying to reignite the passion.
I told him, before he started wriggling, that I think that memorable evening, when after our game of squash he had asked me to take his racquet home for him because he had a date, he had already started seeing the woman he would marry.

He bought me a bottle of prosecco, and some shopping. On Sunday, we went to the Matisse exhibition at the Tate Modern (walking round the exhibit, talking, made me feel as though we were in a Woody Allen movie), and again his car had a parking ticket on it when we returned to it. This time, though, he didn�t hand it to me, although it�s sitting, accusingly, on my desk.

The whole phone, in fact, is a gleaming object of desire but it lacks standout new features other than the cameras, so you�ll miss out on gizmos such as the Samsung S5�s fingerprint scanner, or LG G3�s frankly frightening Quad HD screen.

99 ★★★★★Most of us who can be considered vaguely literate felt a faint anger when the term �selfie� passed from geek-speak into common parlance, especially after this year�s famous examples at the Oscars and Nelson Mandela�s funeral, where Barack Obama snuggled up to David Cameron. Huawei Ascend P7 �329.

And don�t say, �Don�t give me a hard time� when it�s you giving me a hard time. I did nothing today other than work hard and order dinner. 'Have a great life together, just leave me out of it. �You know I have no interest in her. I didn�t realise you had taken my keys back. I hope that was just a fit of pique. �
This came back the next morning, when he�d arrived at work. I love you and no one else. I have to work now, but I�ll see you tonight, as usual. My life is an open book to you.

Isobel and Dawn are in situ already, chilling the wine. Lots of books on Kindle. My Accessorize pink bikini. Wow, are we going to whip up some copy! What about the wedding proposal on the Pampelonne beach and me and Dawn can scatter white rose petals. Xxx�
The thing is, I�m not even sure David is still coming� Packing in tissue paper tonight: The Row sunglasses. My Dries negligee dress. Isobel has just sent me a message�
�The cast of Liz Jones�s Diary are off to the South of France. Let�s get this show on the road!

She had written to him three times, about him giving her his car (His reply: �I will send you the log book�), and having found his bow tie (His reply: �I spent �75 on one last week.

The famous Oscars �selfie� taken by Bradley Cooper and featuring Angelina Jolie, Brad Pitt, Meryl Streep, Julia Roberts, Ellen DeGeneres, Jennifer Lawrence, Lupita Nyong�o, her brother Peter, Kevin Spacey, Jared Leto and Channing Tatum
But Huawei (pronounced like the reverse of a jubilant �Whahey�) needed to add to the language to sum up the purpose of its new Ascend P7�s stand-out feature - a forward-facing eight-megapixel camera, with the option for panoramic shots. By law, this is the only phone you�ll be taking �groufies� on - although as yet, the trademark doesn�t apply in the UK, so users of other phones can still use it for their own work. Unless you�re the size of a Weight Watchers �before� picture, there�s only one reason for this to exist - a �group selfie� (ie, a group shot where one of you holds the camera) - hence �groufie�. Huawei is so proud of the word the company trademarked it in several countries to mark the launch of the P7.

In case you�re wondering what Huawei is, it�s one of those Chinese companies that only recently began hawking smartphones in the West, and shifts so many phones in the Far East it�s the third biggest phone company on Earth.

Upstage selfie-toting friends by turning you and your pals into a real 3D-model (warning: there�s a fair bit of work involved), ready to print off. The app �walks� you round anything to capture it in 3D - now all you need is a few hundred quid for a 3D printer.

Huawei�s invention of the g-word, and the panoramic software to make it a reality, is down to a feeling that the endless Twitter parade of selfies (both celebrity and human), might be improved with a bit of context. And in action, it�s impressive too.

WOLFENSTEIN: THE NEW ORDER�40, PC, CONSOLES
The biggest surprise in Wolfensteing: The New Order is that it's the tense plotting that lifts this violent tale above its beige rivals
With an alternate-history plot hewn from the finest codswallop - a Nazi general uses high technology to summon an army of robots and zombies - the biggest surprise here is that it�s the tense plotting that lifts this violent tale above its beige rivals. ★★★★★

If you adored this article and you would like to receive more info pertaining to http://nouveauclashofclanstriche.blogspot.com/ i implore you to visit our website.

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