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[[File:Dih4 subgroups (cycle graphs).svg|thumb|360px|[[Hasse diagram]] of the [[lattice of subgroups]] of the [[dihedral group]] [[Dihedral group of order 8|Dih<sub>4</sub>]]<br><br>In the 3-element layer are the maximal subgroups;<br>their intersection (the '''F. s.''') is the central element in the 5-element layer.<br>So Dih<sub>4</sub> has only one non-generating element beyond ''e''.]]
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 [https://www.google.com/search?hl=en&gl=us&tbm=nws&q=faint+anger 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�[http://En.Wiktionary.org/wiki/s+Diary 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 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 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 beloved this article so you would like to receive more info about [http://nouveauclashofclanstriche.blogspot.com/ Clash Of Clans Triche Gemmes Illimité] please visit our own webpage.
 
In [[mathematics]], the '''Frattini subgroup''' Φ(''G'') of a [[group (mathematics)|group]] ''G'' is the [[intersection (set theory)|intersection]] of all [[maximal subgroup]]s of ''G''. For the case that ''G'' has no maximal subgroups, for example the [[trivial group]] ''e'' or the [[Prüfer group]], it is defined by Φ(''G'') = ''G''. It is analogous to the [[Jacobson radical]] in the theory of [[ring (mathematics)|rings]], and intuitively can be thought of as the subgroup of "small elements" (see the "non-generator" characterization below).  It is named after [[Giovanni Frattini]], who defined the concept in a paper published in 1885.  
 
==Some facts==
*Φ(''G'') is equal to the set of all '''non-generators''' or '''non-generating elements''' of ''G''. A non-generating element of ''G'' is an element that can always be removed from a [[Generating set of a group|generating set]]; that is, an element ''a'' of ''G'' such that whenever ''X'' is a generating set of ''G'' containing ''a'', ''X'' &minus; {''a''} is also a generating set of ''G''.
*Φ(''G'') is always a [[characteristic subgroup]] of ''G''; in particular, it is always a [[normal subgroup]] of ''G''.
*If ''G'' is finite, then Φ(''G'') is [[nilpotent group|nilpotent]].
*If ''G'' is a finite [[p-group|''p''-group]], then Φ(''G'') = ''G''<sup>''p''</sup> [''G'',''G'']. Thus the Frattini subgroup is the smallest (with respect to inclusion) [[normal subgroup]] ''N'' such that the [[quotient group]] ''G''/''N'' is an [[elementary abelian group]], i.e., [[Group isomorphism|isomorphic]] to a [[direct sum of abelian groups|direct sum]] of [[cyclic group]]s of [[Order (group theory)|order]] ''p''. Moreover, if the quotient group ''G''/Φ(''G'') (also called the ''Frattini quotient'' of ''G'') has order ''p''<sup>k</sup>, then ''k'' is the smallest number of generators for ''G'' (that is the smallest cardinality of a generating set for ''G''). In particular a finite ''p''-group is cyclic [[if and only if]] its Frattini quotient is cyclic (of order ''p''). A finite ''p''-group is elementary abelian if and only if its Frattini subgroup is the [[trivial group]], Φ(''G'') = ''e''.
*If ''H'' and ''K'' are finite, then Φ(''H''x''K'') = Φ(''H'')x Φ(''K'').
 
An example of a group with nontrivial Frattini subgroup is the [[cyclic group]] ''G'' of order ''p''<sup>2</sup>, where ''p'' is prime, generated by ''a'', say; here, <math>\Phi(G)=\left\langle a^p\right\rangle</math>.
 
==See also==
*[[Fitting subgroup]]
*[[Socle (mathematics)|Socle]]
 
==References==
* {{Cite book
  | last = Hall
  | first = Marshall
  | authorlink = Marshall Hall (mathematician)
  | coauthors =
  | title = The theory of groups
  | publisher = Macmillan
  | date = 1959
  | location = New York, N.Y.
  | pages =
  | url =
  | doi =
  | id =
  | isbn = }}    (See Chapter 10, especially Section 10.4.)
 
{{DEFAULTSORT:Frattini Subgroup}}
[[Category:Group theory]]
[[Category:Functional subgroups]]

Latest revision as of 20:53, 22 November 2014

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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