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| In [[mathematics]], especially in the area of [[algebra]] known as [[group theory]], the term '''Z-group''' refers to a number of distinct types of [[group (mathematics)|groups]]:
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| * in the study of [[finite group]]s, a '''Z-group''' is a finite groups whose [[Sylow subgroup]]s are all [[cyclic group|cyclic]].
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| * in the study of [[infinite group]]s, a '''Z-group''' is a group which possesses a very general form of [[central series]].
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| * occasionally, '''(Z)-group''' is used to mean a [[Zassenhaus group]], a special type of [[permutation group]].
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| ==Groups whose Sylow subgroups are cyclic==
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| :''Usage: {{harv|Suzuki|1955}}, {{harv|Bender|Glauberman|1994|p=2}}, {{MR|0409648}}, {{harv|Wonenburger|1976}}, {{harv|Çelik|1976}}''
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| In the study of [[finite group]]s, a '''Z-group''' is a finite group whose [[Sylow subgroup]]s are all [[cyclic group|cyclic]]. The Z originates both from the German [[:de:Zyclische gruppe|''Zyklische'']] and from their classification in {{harv|Zassenhaus|1935}}. In many standard textbooks<!-- burnside, huppert, gorenstein, robinson --> these groups have no special name, other than '''metacyclic groups''', but that term is often<!-- huppert, gorenstein, robinson --> used more generally today. See [[metacyclic group]] for more on the general, modern definition which includes non-cyclic [[p-group|''p''-groups]]; see {{harv|Hall|1969|loc=Th. 9.4.3}} for the stricter, classical definition more closely related to Z-groups.
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| Every group whose Sylow subgroups are cyclic is itself [[metacyclic group|metacyclic]], so [[supersolvable group|supersolvable]]. In fact, such a group has a cyclic [[derived subgroup]] with cyclic maximal abelian quotient. Such a group has the presentation {{harv|Hall|1969|loc=Th. 9.4.3}}:
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| :<math>G(m,n,r) = \langle a,b | a^n = b^m = 1, a^b = a^r \rangle </math>, where ''mn'' is the order of ''G''(''m'',''n'',''r''), the [[greatest common divisor]], gcd((''r''-1)''n'', ''m'') = 1, and ''r''<sup>''n''</sup> ≡ 1 (mod ''m'').
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| The [[character theory]] of Z-groups is well understood {{harv|Çelik|1976}}, as they are [[monomial group]]s.
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| The derived length of a Z-group is at most 2, so Z-groups may be insufficient for some uses. A generalization due to Hall are the [[A-group]]s, those groups with [[abelian group|abelian]] Sylow subgroups. These groups behave similarly to Z-groups, but can have arbitrarily large derived length {{harv|Hall|1940}}. Another generalization due to {{harv|Suzuki|1955}} allows the Sylow 2-subgroup more flexibility, including [[dihedral group|dihedral]] and [[generalized quaternion group]]s.
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| ==Group with a generalized central series==
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| :''Usage: {{harv|Robinson|1996}}, {{harv|Kurosh|1960}}''
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| The definition of [[central series]] used for '''Z-group''' is somewhat technical. A '''series''' of ''G'' is a collection ''S'' of subgroups of ''G'', linearly ordered by inclusion, such that for every ''g'' in ''G'', the subgroups ''A''<sub>''g''</sub> = ∩ { ''N'' in ''S'' : ''g'' in ''N'' } and ''B''<sub>''g''</sub> = ∪ { ''N'' in ''S'' : ''g'' not in ''N'' } are both in ''S''. A (generalized) '''central series''' of ''G'' is a series such that every ''N'' in ''S'' is normal in ''G'' and such that for every ''g'' in ''G'', the quotient ''A''<sub>''g''</sub>/''B''<sub>''g''</sub> is contained in the center of ''G''/''B''<sub>''g''</sub>. A '''Z'''-group is a group with such a (generalized) central series. Examples include the [[hypercentral group]]s whose transfinite [[upper central series]] form such a central series, as well as the [[hypocentral group]]s whose transfinite lower central series form such a central series {{harv|Robinson|1996}}.
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| ==Special 2-transitive groups==
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| {{main|Zassenhaus group}}
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| :''Usage: {{harv|Suzuki|1961}}''
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| A '''(Z)-group''' is a group faithfully represented as a [[doubly transitive permutation group]] in which no non-identity element fixes more than two points. A '''(ZT)-group''' is a (Z)-group that is of odd degree and not a [[Frobenius group]], that is a [[Zassenhaus group]] of odd degree, also known as one of the groups [[projective special linear group|PSL(2,2<sup>''k''+1</sup>)]] or [[Group of Lie type#Suzuki–Ree groups|Sz(2<sup>2''k''+1</sup>)]], for ''k'' any positive integer {{harv|Suzuki|1961}}.
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| ==References==
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| *{{Citation | last1=Bender | first1=Helmut | last2=Glauberman | first2=George | author2-link=George Glauberman | title=Local analysis for the odd order theorem | publisher=[[Cambridge University Press]] | series=London Mathematical Society Lecture Note Series | isbn=978-0-521-45716-3 | mr=1311244 | year=1994 | volume=188}}
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| *{{Citation | last1=Çelik | first1=Özdem | title=On the character table of Z-groups | mr=0470050 | year=1976 | journal=Mitteilungen aus dem Mathematischen Seminar Giessen | issn=0373-8221 | pages=75–77}}
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| * {{citation | last=Hall, jr | first=Marshall | authorlink=Marshall Hall (mathematician) | title=The Theory of Groups | year=1969 | publisher=Macmillan | location=New York }}
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| *{{Citation | last1=Hall | first1=Philip | author1-link=Philip Hall | title=The construction of soluble groups | mr=0002877 | year=1940 | journal=Journal für die reine und angewandte Mathematik | issn=0075-4102 | volume=182 | pages=206–214}}
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| *{{Citation | last1=Kurosh | first1=A. G. | title=The theory of groups | publisher=Chelsea | location=New York | mr=0109842 | year=1960}}
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| *{{Citation | last1=Robinson | first1=Derek John Scott | title=A course in the theory of groups | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-94461-6 | year=1996}}
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| *{{Citation | last1=Suzuki | first1=Michio | author1-link=Michio Suzuki | title=On finite groups with cyclic Sylow subgroups for all odd primes | mr=0074411 | year=1955 | journal=[[American Journal of Mathematics]] | issn=0002-9327 | volume=77 | pages=657–691 | doi=10.2307/2372591 | jstor=2372591 | issue=4}}
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| *{{Citation | last1=Suzuki | first1=Michio | author1-link=Michio Suzuki | title=Finite groups with nilpotent centralizers | mr=0131459 | year=1961 | journal=[[Transactions of the American Mathematical Society]] | issn=0002-9947 | volume=99 | pages=425–470 | doi=10.2307/1993556 | jstor=1993556 | issue=3}}
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| *{{Citation | last1=Wonenburger | first1=María J. | title=A generalization of Z-groups | mr=0393229 | year=1976 | journal=Journal of Algebra | issn=0021-8693 | volume=38 | issue=2 | pages=274–279 | doi=10.1016/0021-8693(76)90219-2}}
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| *{{Citation | last1=Zassenhaus | first1=Hans | author1-link=Hans Zassenhaus | title=Über endliche Fastkörper | language=German | year=1935 | journal=Abh. Math. Semin. Hamb. Univ. | volume=11 | pages=187–220 | doi=10.1007/BF02940723}}
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| [[Category:Infinite group theory]]
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| [[Category:Finite groups]]
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| [[Category:Properties of groups]]
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We all know how EASY it is to start whole new diet. The excitement of finding something new. The hope than me working for us and doing exactly what we want it to do. Whether it starts as a new year's resolution, or something in the middle of a random week, is undoubtedly always the chance that Affliction be the diet that will transform us.
The server rung up my order and requested that I hand over seven dollars and change else my waffle platter would remain hostage inside kitchen. I sheepishly complied and she handed inside the goods. Upon first glance of my meal There we were immediately sorry. First of all, I could not believe which i paid $7 for this skimpy little plate of food. The Belgian waffle somehow didn't appear "Belgian enough" and also the two side items looked barely done; it's probably never some sort of idea to consume rare bacon and cash. Belgian waffles are usually crispier, and sweeter than American waffles, so much so that It's my job to never need to put yacon syrup reviews on a Belgian waffle.
WHOLE GRAINS: Whole grains are rich in fiber, which acts becoming a speed bump in your gastrointestinal tract, slowing everything down, that means you feel fuller longer.
Breakfast may be the most important meal for the day, and still not to be skipped, as it jump starts the practice. Dr. Roizen suggests some monotony for breakfast and lunch, to eliminate having to produce choices Good breakfast choices are: steel-cut oatmeal, wholemeal cereals and egg-white omelets.
Have you ever eaten shelled sunflower seeds? Then you be aware they are slightly oily and contain fat. Concerning that same sunflower seed 10 days later after it's sprouted into a sunflower blue? Now it's green crunchy, fresh and plump with water. Where did body fat go? Merge of sprouting changed the nutritional composition of the seed.
Finally do not try to achieve this alone. There a great free video that explains all advisors fat burning concepts too as how you can plan out tasty meals considering your exact metabolic profile. This will give everybody the give you support need to get rid of the weight you i never thought possible.
First, get the jello and placed it setting and chill. Next, put cottage cheese in a medium bowl and add the blueberries from the can (after draining it) to this task. Mix carefully and then add some food coloring. For serving, put some Jello on a plate, sprinkle some of this blueberry syrup on it (a fair amount) which usually place the cheesy blueberry mixture upon. Let your attendees drool inside the gross main course.