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| In [[mathematics]], especially [[group theory]], the elements of any [[group (mathematics)|group]] may be [[partition of a set|partitioned]] into '''conjugacy classes'''; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure.<ref>{{cite book | last1=Dummit | first1=David S. | last2=Foote | first2=Richard M. | title=Abstract Algebra | publisher=[[John Wiley & Sons]] | year=2004 | edition=3rd | isbn=0-471-43334-9}}</ref><ref>{{cite book | last=Lang | first=Serge | authorlink=Serge Lang | title=Algebra | publisher=[[Springer Science+Business Media|Springer]] | series=[[Graduate Texts in Mathematics]] | year=2002 | isbn=0-387-95385-X}}</ref> In all [[abelian group]]s every conjugacy class is a set containing one element ([[Singleton (mathematics)|singleton set]]).
| | 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> |
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| [[function (mathematics)|Function]]s that are constant for members of the same conjugacy class are called [[class function]]s.
| | 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 ★★★★★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 [http://www.Answers.com/topic/purpose 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 3D 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 [http://www.dict.cc/?s=technology 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. ★★★★★<br><br>In the event you cherished this information along with you would like to receive more details regarding [http://nouveauclashofclanstriche.Blogspot.com/ http://nouveauclashofclanstriche.blogspot.com/] generously go to our site. |
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| ==Definition==
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| Suppose ''G'' is a group. Two elements ''a'' and ''b'' of ''G'' are called '''conjugate''' if there exists an element ''g'' in ''G'' with
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| :''gag''<sup>−1</sup> = ''b''.
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| (In [[linear algebra]], this is referred to as [[matrix similarity]].)
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| It can be readily shown that conjugacy is an [[equivalence relation]] and therefore partitions ''G'' into [[equivalence class]]es. (This means that every element of the group belongs to precisely one conjugacy class, and the classes Cl(''a'') and Cl(''b'') are equal [[if and only if]] ''a'' and ''b'' are conjugate, and [[Disjoint sets|disjoint]] otherwise.) The equivalence class that contains the element ''a'' in ''G'' is
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| :Cl(''a'') = { ''gag''<sup>−1</sup>: ''g'' ∈ ''G'' }
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| and is called the '''conjugacy class''' of ''a''. The '''class number''' of ''G'' is the number of distinct (nonequivalent) conjugacy classes. All elements belonging to the same conjugacy class have the same [[Order (group theory)|order]].
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| Conjugacy classes may be referred to by describing them, or more briefly by abbreviations such as "6A", meaning "a certain conjugacy class of order 6 elements", and "6B" would be a different conjugacy class of order 6 elements; the conjugacy class 1A is the conjugacy class of the identity. In some cases, conjugacy classes can be described in a uniform way – for example, in the symmetric group they can be described by cycle structure.
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| ==Examples==
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| The [[symmetric group]] ''S''<sub>3</sub>, consisting of all 6 [[permutation]]s of three elements, has three conjugacy classes:
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| *no change (abc → abc)
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| *interchanging two (abc → acb, abc → bac, abc → cba)
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| *a cyclic permutation of all three (abc → bca, abc → cab)
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| The symmetric group ''S''<sub>4</sub>, consisting of all 24 permutations of four elements, has five conjugacy classes, listed with their cycle structures and orders:
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| *(1)<sub>4</sub>: no change (1 element: { {1, 2, 3, 4} } )
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| *(2): interchanging two (6 elements: { {1, 2, 4, 3}, {1, 4, 3, 2}, {1, 3, 2, 4}, {4, 2, 3, 1}, {3, 2, 1, 4}, {2, 1, 3, 4} })
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| *(3): a cyclic permutation of three (8 elements: { {1, 3, 4, 2}, {1, 4, 2, 3}, {3, 2, 4, 1}, {4, 2, 1, 3}, {4, 1, 3, 2}, {2, 4, 3, 1}, {3, 1, 2, 4}, {2, 3, 1, 4} } )
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| *(4): a cyclic permutation of all four (6 elements: { {2, 3, 4, 1}, {2, 4, 1, 3}, {3, 1, 4, 2}, {3, 4, 2, 1}, {4, 1, 2, 3}, {4, 3, 1, 2} } )
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| *(2)(2): interchanging two, and also the other two (3 elements: { {2, 1, 4, 3}, {4, 3, 2, 1}, {3, 4, 1, 2} } )
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| In general, the number of conjugacy classes in the symmetric group ''S''<sub>n</sub> is equal to the number of [[integer partition]]s of ''n''. This is because each conjugacy class corresponds to exactly one partition of {1, 2, ..., ''n''} into [[cycle notation|cycles]], up to permutation of the elements of {1, 2, ..., ''n''}.
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| See also the [[Octahedral symmetry#The isometries of the cube|proper rotations of the cube]], which can be characterized by permutations of the body diagonals.
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| ==Properties==
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| * The identity element is always in its own class, that is Cl(''e'') = {''e''}
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| * If ''G'' is [[abelian group|abelian]], then ''gag''<sup>−1</sup> = ''a'' for all ''a'' and ''g'' in ''G''; so Cl(''a'') = {''a''} for all ''a'' in ''G''; the concept is therefore not very useful in the abelian case. The failure of this thus gives us an idea in what degree the group is nonabelian.
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| * If two elements ''a'' and ''b'' of ''G'' belong to the same conjugacy class (i.e., if they are conjugate), then they have the same [[order (group theory)|order]]. More generally, every statement about ''a'' can be translated into a statement about ''b''=''gag''<sup>−1</sup>, because the map φ(''x'') = ''gxg''<sup>−1</sup> is an [[group isomorphism#Automorphisms|automorphism]] of ''G''.
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| * An element ''a'' of ''G'' lies in the [[center of a group|center]] Z(''G'') of ''G'' if and only if its conjugacy class has only one element, ''a'' itself. More generally, if C<sub>''G''</sub>(''a'') denotes the ''[[centralizer]]'' of ''a'' in ''G'', i.e., the [[subgroup]] consisting of all elements ''g'' such that ''ga'' = ''ag'', then the [[index of a subgroup|index]] [''G'' : C<sub>''G''</sub>(''a'')] is equal to the number of elements in the conjugacy class of ''a'' (by the [[orbit-stabilizer theorem]]).
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| * If ''a'' and ''b'' are conjugate, then so are powers of them, <math>a^k</math> and <math>b^k</math> – thus taking ''k''th powers gives a map on conjugacy classes, and one may speak of which conjugacy classes which are in the preimage of this map. For example, in the symmetric group, the square of an element of type (3)(2) (a 3-cycle and a 2-cycle) is an element of type (3), therefore one of the power-up classes of (3) is the class (3)(2); the class (6) is another.
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| ==Conjugacy class equation==
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| If ''G'' is a [[finite group]], then for any group element ''a'', the elements in the conjugacy class of ''a'' are in one-to-one correspondence with [[coset]]s of the [[centralizer]] ''C''<sub>''G''</sub>(''a''). This can be seen by observing that any two elements ''b'' and ''c'' belonging to the same coset (and hence, ''b''=''cz'' for some ''z'' in the centralizer ''C''<sub>''G''</sub>(''a'')) give rise to the same element when conjugating ''a'': ''bab''<sup>−1</sup>=''cza''(''cz'')<sup>−1</sup>=''czaz''<sup>−1</sup>''c''<sup>−1</sup>=''czz''<sup>−1</sup>''ac''<sup>−1</sup>=''cac''<sup>−1</sup>.
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| Thus the number of elements in the conjugacy class of ''a'' is the [[index of a subgroup|index]] [''G'':''C''<sub>''G''</sub>(''a'')] of the centralizer ''C''<sub>''G''</sub>(''a'') in ''G''. Thus the size of each conjugacy class is a divisor of the order of the group.
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| Furthermore, if we choose a single representative element ''x''<sub>''i''</sub> from every conjugacy class, we infer from the disjointedness of the conjugacy classes that |''G''| = ∑<sub>''i''</sub> [''G'' : ''C''<sub>''G''</sub>(''x''<sub>''i''</sub>)], where ''C''<sub>''G''</sub>(''x''<sub>''i''</sub>) is the centralizer of the element ''x''<sub>''i''</sub>. Observing that each element of the center Z(''G'') forms a conjugacy class containing just itself gives rise to the following important '''class equation''':<ref>Grillet (2007), {{Google books quote|id=LJtyhu8-xYwC|page=57|text=The Class Equation|p. 57}}</ref>
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| :|''G''| = |Z(''G'')| + ∑<sub>''i''</sub> [''G'' : ''C''<sub>''G''</sub>(''x''<sub>''i''</sub>)]
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| where the second sum is over a representative element from each conjugacy class that is not in the center.
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| Knowledge of the divisors of the group order |''G''| can often be used to gain information about the order of the center or of the conjugacy classes.
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| ===Example===
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| Consider a finite [[p-group|''p''-group]] ''G'' (that is, a group with order ''p''<sup>''n''</sup>, where ''p'' is a [[prime number]] and ''n'' > 0). We are going to prove that: ''every finite ''p''-group has a non-[[Trivial (mathematics)|trivial]] center''.
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| Since the order of any conjugacy class of ''G'' must divide the order of ''G'', it follows that each conjugacy class ''H''<sub>''i''</sub> also has order some power of ''p''<sup>''k''<sub>''i''</sub></sup>, where 0 < ''k''<sub>''i''</sub> < ''n''. But then the class equation requires that |''G''| = ''p''<sup>''n''</sup> = |Z(''G'')| + ∑<sub>''i''</sub> ''p''<sup>''k''<sub>''i''</sub></sup>. From this we see that ''p'' must divide |Z(''G'')|, so |Z(''G'')| > 1.
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| ==Conjugacy of subgroups and general subsets==<!-- This section is linked from [[Symmetry group]] -->
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| More generally, given any [[subset]] ''S'' of ''G'' (''S'' not necessarily a subgroup), we define a subset ''T'' of ''G'' to be conjugate to ''S'' if there exists some ''g'' in ''G'' such that ''T'' = ''gSg''<sup>−1</sup>. We can define '''Cl(''S'')''' as the set of all subsets ''T'' of ''G'' such that ''T'' is conjugate to ''S''.
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| A frequently used theorem is that, given any subset ''S'' of ''G'', the [[coset|index]] of N(''S'') (the [[normalizer]] of ''S'') in ''G'' equals the order of Cl(''S''):
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| :|Cl(''S'')| = [''G'' : N(''S'')]
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| This follows since, if ''g'' and ''h'' are in ''G'', then ''gSg''<sup>−1</sup> = ''hSh''<sup>−1</sup> if and only if ''g''<sup>−1</sup>''h'' is in N(''S''), in other words, if and only if ''g'' and ''h'' are in the same [[coset]] of N(''S'').
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| Note that this formula generalizes the one given earlier for the number of elements in a conjugacy class (let ''S'' = {''a''}).
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| The above is particularly useful when talking about subgroups of ''G''. The subgroups can thus be divided into conjugacy classes, with two subgroups belonging to the same class if and only if they are conjugate.
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| Conjugate subgroups are [[group isomorphism|isomorphic]], but isomorphic subgroups need not be conjugate. For example, an abelian group may have two different subgroups which are isomorphic, but they are never conjugate.
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| ==Conjugacy as group action==<!-- This section is linked from [[Quaternions and spatial rotation]] -->
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| If we define
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| :''g . x'' = ''gxg''<sup>−1</sup> | |
| for any two elements ''g'' and ''x'' in ''G'', then we have a [[group action]] of ''G'' on ''G''. The [[Group_action#Orbits_and_stabilizers|orbits]] of this action are the conjugacy classes, and the [[Group_action#Orbits_and_stabilizers|stabilizer]] of a given element is the element's [[centralizer]].<ref name="Grillet-2007-p56">Grillet (2007), {{Google books quote|id=LJtyhu8-xYwC|page=56|text=the orbits are the conjugacy classes|p. 56}}</ref>
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| Similarly, we can define a group action of ''G'' on the set of all subsets of ''G'', by writing
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| :''g . S'' = ''gSg''<sup>−1</sup>, | |
| or on the set of the subgroups of ''G''.
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| ==Geometric interpretation==
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| Conjugacy classes in the [[fundamental group]] of a [[path-connected]] topological space can be thought of as equivalence classes of [[free loop]]s under free homotopy.
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| ==See also==
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| * [[Topological conjugacy]]
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| * [[FC-group]]
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| ==References==
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| {{reflist}}
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| * {{cite book |last1=Grillet |first1=Pierre Antoine |authorlink1= |last2= |first2= |authorlink2= |title=Abstract algebra |url= |edition=2 |series=Graduate texts in mathematics |volume=242 |year=2007 |publisher=Springer |location= |isbn=978-0-387-71567-4 |id= }}
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| [[Category:Group theory]]
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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. ★★★★★
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