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	~~{{DISPLAYTITLE:Powerful ''p''-group}}~~		William Lagakos confirms that Wolf is correct in his analysis. The diet is also called the Cave Man Diet, Nutrition secrets can be learned from the caveman diet because the modern day diet is deficient in healthy foods to eat. Just because smoothies are created from natural basic elements, the chemical preservatives are way reduced also. Eat the original diet designed for optimal health, strength and longevity. Sisson said he improved his athletic performance, his body composition, and his energy levels after switching to a low-carb, high-fat (LCHF) Paleo diet. <br><br>Now many people are following the paleo diet which is also called the stone age, palaeolithic, primal, caveman or low carbohydrate diet (though it's different to the atkins diet). You can also enjoy an omelet created using two eggs which contains newly harvested produce such as peppers, tomatoes, onions, and mushrooms You may also take pleasure in treats of fruits as well as veggies including all types of berries, uncooked carrots, as well as celery. The Paleo diet goes back to the time when humans were still a nomadic people, before they settled down into cities and relied on agriculture to provide them with food. The paleo diet greatly encourages those that are following it to consume more vegetables. Evolution has not caught up with our present day millisecond mentality. <br><br>The reason for this is because the cows are free to roam around and eat all the grass. "I saw the difference in my body, my energy and my mental clarity. Both fruit and veggies contain a number of vitamins. Today, it appears to be a staple of the American diet. In spite of this, there are numerous foods included on the list for this diet that are both perfect for satisfying snack cravings and for using in recipes. <br><br>Reduce the amount of salt or increase the volume of the water. The Paleo diet has additional food restrictions that encourages optimal long term health. Desserts are usually made from fruits such as apples and cranberries. Also the time frame in which you lose weight is typically much longer than that of the 24 Day Challenge. Results from most diets take a great deal of time, but not with this plan. <br><br>Or at least the leftover carcass of a wolf-pack kill. At that time, we human beings decided to end looking and collecting and start planting. Try drying them out if you can't use all of them or buy some of the potted plants that you can keep for a long time. Evenings: Utilize what is known as the oil cleansing method. Heart disease and stroke is the leading cause of death and disability in America and the prime reason behind this is the sedentary lifestyle and a diet high in calories and saturated fat. <br><br>Also, the diet is rather extreme and will make most people feel severely restricted and even stifled. Polyunsaturated fats (good fats) help reduce blood cholesterol and reduce risk of heart diseases. It is referred to as the Stone Age, Caveman or Hunter-Gatherer diet. You know, the Mediterranean diet is considered to be one of the healthiest diets in the world today. Some even documented which they were capable of do away with chronic health conditions.<br><br>If you treasured this article therefore you would like to collect more info concerning [http://gritsandgroceries.info/ does the paleo diet work] please visit our site.
	~~In [[mathematics]],~~ in the ~~field of [[group theory]]~~, ~~especially in~~ the ~~study of [[p-group\|''p''-groups]] and [[pro-p-group\|pro-''p''-groups]],~~ the ~~concept of '''powerful ''p''-groups''' plays an important role~~. ~~They were introduced in {{harv\|Lubotzky\|Mann\|1987}}~~, ~~where a number of applications~~ are ~~given, including results on [[Schur multiplier]]s~~. ~~Powerful ''p''-groups are used in~~ the ~~study of [[automorphism]]s of ''p''-groups {{harv\|Khukhro\|1998}}~~, ~~the solution of the [[restricted Burnside problem]] {{harv\|Vaughan-Lee\|1993}}~~, ~~the classification of finite ''p''-groups via the coclass conjectures {{harv\|Leedham-Green\|McKay\|2002}}~~, and ~~provided an excellent method of understanding analytic pro~~-~~''p''~~-~~groups {{harv\|Dixon\|du Sautoy\|Mann\|Segal\|1991}}~~.

	~~==Formal definition==~~
	~~A finite ''p''-group~~ <~~math~~>G<~~/math~~> is called '~~''powerful''' if~~ the ~~[[commutator subgroup]] <math>[G~~,~~G]</math> is contained~~ in the ~~subgroup~~ <~~math~~>~~G^p = \langle g^p \| g\in G\rangle~~<~~/math~~> for ~~odd <math>p</math>, or if <math>[G,G]</math>~~ is ~~contained in~~ the ~~subgroup <math>G^4</math> for ''p''=2~~.

	~~==Properties of powerful ''p''-groups==~~
	~~Powerful ''p''-groups have many properties similar to [[abelian group]]s~~, and ~~thus provide~~ a ~~good basis for studying ''p''-groups~~. ~~Every finite ''p''-group can~~ be ~~expressed as~~ a ~~[[section (group theory)\|section]]~~ of ~~a powerful ''p''-group~~.

	~~Powerful ''p''-groups~~ are ~~also useful in~~ the ~~study of [[pro-p group\|pro-''p'' groups]] as it provides a simple means~~ for ~~characterising [[p-adic number\|''p''-adic]] analytic groups (groups~~ that are ~~[[manifold]]s over the ''p''-adic numbers): A finitely generated pro-''p'' group is ''p''-adic analytic if~~ and ~~only if it contains an [[Open set\|open]] [[normal subgroup]] that is powerful: this is a special case of a deep result of [[Michel Lazard]] (1965)~~.

	~~Some properties similar to abelian [[p-group\|''p''-groups]] are: if~~ <~~math~~>G<~~/math~~> ~~is a powerful ''p''-group then:~~
	* The ~~[[Frattini subgroup]] <math>\Phi(G)</math> of <math>G</math>~~ has the ~~property <math>\Phi(G) = G^p.</math>~~
	* <math>G^{p^k} = \{g^{p^k}\|g\in ~~G\}</math> for all <math>k\geq 1~~.~~</math> That is~~, ~~the ''group generated'' by~~ <~~math~~>p<~~/math~~>~~th powers is precisely~~ the ~~''set''~~ of ~~<math>p</math>th powers~~.
	* If <math>G = \langle g_1, ~~\ldots, g_d\rangle</math> then <math>G^{p^k} = \langle g_1^{p^k},\ldots,g_d^{p^k}\rangle</math> for~~ all ~~<math>k\geq 1.</math>~~
	* The <math>k</math>th entry of ~~the [[lower central series]]~~ of ~~<math>G</math> has~~ the ~~property <math>\gamma_k(G)\leq G^{p^{k-1}}</math>~~ for ~~all <math>k\geq 1~~.~~</math>~~
	* Every [[quotient group]] of a powerful ''p''-group is ~~powerful~~.
	* The [[Prüfer rank]] of <math>G</math> is ~~equal to~~ the ~~minimal number~~ of ~~generators of <math>G.</math>~~

	~~Some less abelian-like properties are: if <math>G</math>~~ is a ~~powerful ''p''-group then:~~
	* <~~math~~>~~G^{p^k}~~<~~/math~~> is ~~powerful~~.
	* [[Subgroups]] of ~~<math>G</math> are not necessarily powerful~~.

	~~==References==~~
	* Lazard, ~~Michel (1965), Groupes analytiques p~~-~~adiques, Publ~~.~~Math.IHES 26 (1965)~~, ~~389-603~~.
	* {{citation \| id={{MathSciNet \| id=1152800}} \| last1=Dixon \| first1=J. D. \| last2=du Sautoy \| first2=M. P. F. \| author2-link=Marcus du Sautoy \| last3=Mann \| first3=A. \| last4=Segal \| first4=D. \| title=Analytic pro-p-groups \| publisher=[[Cambridge University Press]] \| year=1991 \| isbn=0-521-39580-1}}
	* {{citation \| id={{MathSciNet \| id=1615819}} \| last1=Khukhro \| first1=E. I. \| title=p-automorphisms of ~~finite p-groups \| publisher=[[Cambridge University Press]] \| year=1998 \| isbn=0-521-59717-X}}~~
	*{{Citation \| last1=Leedham-Green \| first1=C. ~~R. \| author1-link=Charles Leedham-Green \| last2=McKay \| first2=Susan \| title=The structure of groups of prime power order \| publisher=~~[~~[Oxford University Press]] \| series=London Mathematical Society Monographs. New Series \| isbn=978-0-19-853548-5 \| id={{MathSciNet \| id = 1918951}} \| year=2002 \| volume=27}}~~
	* {{citation \| id={{MathSciNet \| id=0873681}} \| last1=Lubotzky \| author1-link = Alexander Lubotzky \| first1=Alexander \| last2=Mann \| first2=Avinoam \| title=Powerful p-groups. I. Finite Groups \| journal=J. Algebra \| volume=105 \| year=1987 \| pages=484–505 \| doi=10.~~1016~~/~~0021-8693(87)90211-0 \| issue=2}}~~
	* {{citation \| id={{MathSciNet \| id=1364414}} \| last1=Vaughan-Lee \| first1=Michael \| title=The restricted Burnside problem \| edition=2nd \| publisher=[[Oxford University Press]] \| year=1993 \| isbn=0-19-853786-7 }}

	~~[[Category:Group theory]]~~
	~~[[Category:P-groups]]~~
	~~[[Category:Properties of groups]~~]

en>Chris the speller: /* Example */Typo fixing, replaced: half lives → half-lives, typos fixed: , → , using AWB (7774)

2011-07-14T19:30:52Z

Example: Typo fixing, replaced: half lives → half-lives, typos fixed: , → , using AWB (7774)

New page

{{DISPLAYTITLE:Powerful ''p''-group}}
In [[mathematics]], in the field of [[group theory]], especially in the study of [[p-group|''p''-groups]] and [[pro-p-group|pro-''p''-groups]], the concept of '''powerful ''p''-groups''' plays an important role. They were introduced in {{harv|Lubotzky|Mann|1987}}, where a number of applications are given, including results on [[Schur multiplier]]s. Powerful ''p''-groups are used in the study of [[automorphism]]s of ''p''-groups {{harv|Khukhro|1998}}, the solution of the [[restricted Burnside problem]] {{harv|Vaughan-Lee|1993}}, the classification of finite ''p''-groups via the coclass conjectures {{harv|Leedham-Green|McKay|2002}}, and provided an excellent method of understanding analytic pro-''p''-groups {{harv|Dixon|du Sautoy|Mann|Segal|1991}}.

==Formal definition==
A finite ''p''-group <math>G</math> is called '''powerful''' if the [[commutator subgroup]] <math>[G,G]</math> is contained in the subgroup <math>G^p = \langle g^p | g\in G\rangle</math> for odd <math>p</math>, or if <math>[G,G]</math> is contained in the subgroup <math>G^4</math> for ''p''=2.

==Properties of powerful ''p''-groups==
Powerful ''p''-groups have many properties similar to [[abelian group]]s, and thus provide a good basis for studying ''p''-groups. Every finite ''p''-group can be expressed as a [[section (group theory)|section]] of a powerful ''p''-group.

Powerful ''p''-groups are also useful in the study of [[pro-p group|pro-''p'' groups]] as it provides a simple means for characterising [[p-adic number|''p''-adic]] analytic groups (groups that are [[manifold]]s over the ''p''-adic numbers): A finitely generated pro-''p'' group is ''p''-adic analytic if and only if it contains an [[Open set|open]] [[normal subgroup]] that is powerful: this is a special case of a deep result of [[Michel Lazard]] (1965).

Some properties similar to abelian [[p-group|''p''-groups]] are: if <math>G</math> is a powerful ''p''-group then:
* The [[Frattini subgroup]] <math>\Phi(G)</math> of <math>G</math> has the property <math>\Phi(G) = G^p.</math>
* <math>G^{p^k} = \{g^{p^k}|g\in G\}</math> for all <math>k\geq 1.</math> That is, the ''group generated'' by <math>p</math>th powers is precisely the ''set'' of <math>p</math>th powers.
* If <math>G = \langle g_1, \ldots, g_d\rangle</math> then <math>G^{p^k} = \langle g_1^{p^k},\ldots,g_d^{p^k}\rangle</math> for all <math>k\geq 1.</math>
* The <math>k</math>th entry of the [[lower central series]] of <math>G</math> has the property <math>\gamma_k(G)\leq G^{p^{k-1}}</math> for all <math>k\geq 1.</math>
* Every [[quotient group]] of a powerful ''p''-group is powerful.
* The [[Prüfer rank]] of <math>G</math> is equal to the minimal number of generators of <math>G.</math>

Some less abelian-like properties are: if <math>G</math> is a powerful ''p''-group then:
* <math>G^{p^k}</math> is powerful.
* [[Subgroups]] of <math>G</math> are not necessarily powerful.

==References==
* Lazard, Michel (1965), Groupes analytiques p-adiques, Publ.Math.IHES 26 (1965), 389-603.
* {{citation | id={{MathSciNet | id=1152800}} | last1=Dixon | first1=J. D. | last2=du Sautoy | first2=M. P. F. | author2-link=Marcus du Sautoy | last3=Mann | first3=A. | last4=Segal | first4=D. | title=Analytic pro-p-groups | publisher=[[Cambridge University Press]] | year=1991 | isbn=0-521-39580-1}}
* {{citation | id={{MathSciNet | id=1615819}} | last1=Khukhro | first1=E. I. | title=p-automorphisms of finite p-groups | publisher=[[Cambridge University Press]] | year=1998 | isbn=0-521-59717-X}}
*{{Citation | last1=Leedham-Green | first1=C. R. | author1-link=Charles Leedham-Green | last2=McKay | first2=Susan | title=The structure of groups of prime power order | publisher=[[Oxford University Press]] | series=London Mathematical Society Monographs. New Series | isbn=978-0-19-853548-5 | id={{MathSciNet | id = 1918951}} | year=2002 | volume=27}}
* {{citation | id={{MathSciNet | id=0873681}} | last1=Lubotzky | author1-link = Alexander Lubotzky | first1=Alexander | last2=Mann | first2=Avinoam | title=Powerful p-groups. I. Finite Groups | journal=J. Algebra | volume=105 | year=1987 | pages=484–505 | doi=10.1016/0021-8693(87)90211-0 | issue=2}}
* {{citation | id={{MathSciNet | id=1364414}} | last1=Vaughan-Lee | first1=Michael | title=The restricted Burnside problem | edition=2nd | publisher=[[Oxford University Press]] | year=1993 | isbn=0-19-853786-7 }}

[[Category:Group theory]]
[[Category:P-groups]]
[[Category:Properties of groups]]

Decay correct - Revision history

71.84.176.43 at 19:56, 29 July 2014

en>Chris the speller: /* Example */Typo fixing, replaced: half lives → half-lives, typos fixed: , → , using AWB (7774)