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In [[mathematics]], the '''quasi-dihedral groups''' and '''semi-dihedral groups''' are non-abelian [[group (mathematics)|group]]s of order a power of 2.  For every positive integer n greater than or equal to 4, there are exactly four isomorphism classes of nonabelian groups of order 2<sup>n</sup> which have a cyclic subgroup of [[Index of a subgroup|index]] 2.  Two are well established, the [[generalized quaternion group]] and the [[dihedral group]], but the other two have conflicting names, so this article discusses both groups.


One of the groups is often considered particularly important, since it is an example of a 2-group of [[maximal class|maximal nilpotency class]].  In [[Bertram Huppert]]'s influential text ''Endliche Gruppen'', this group is called a Quasidiedergruppe.  In [[Daniel Gorenstein]]'s influential text, ''Finite Groups'', this group is called the semidihedral group. Dummit and Foote refer to it as the quasidihedral group; we adopt that name in this article. All give the same presentation for this group:


:<math>\langle r,s \mid r^{2^{n-1}} = s^2 = 1, srs = r^{2^{n-2}-1}\rangle\,\!</math>.
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The other non-abelian 2-group with cyclic subgroup of index 2 is not given a special name in either text, but referred to as just G or M_m(2). In this article, this group is called the other-maximal-cyclic group. Dummit and Foote refer to this group as the modular group. Its presentation in both texts is:
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:<math>\langle r,s \mid r^{2^{n-1}} = s^2 = 1, srs = r^{2^{n-2}+1}\rangle\,\!</math>.
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Both these two groups and the dihedral group are [[semidirect product]]s of a cyclic group <''r''> of order 2<sup>n-1</sup> with a cyclic group <''s''> of order 2.  Such a non-abelian semi-direct product is uniquely determined by an element of order 2 in the group of units of the ring <math>\Bbb{Z}/2^{n-1}\Bbb{Z}</math> and there are precisely three such elements, <math>2^{n-1}-1</math>, <math>2^{n-2}-1</math>, and <math>2^{n-2}+1</math>, corresponding to the dihedral group, the quasidihedral, and the other-maximal-cyclic group.
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The generalized quaternion group, the dihedral group, and the quasidihedral group of order 2<sup>''n''</sup> all have nilpotency class ''n''&minus;1, and are the only isomorphism classes of groups of order 2<sup>''n''</sup> with nilpotency class ''n''&minus;1.  The groups of order ''p''<sup>''n''</sup> and nilpotency class ''n''&minus;1 were the beginning of the classification of all [[p-group]]s via [[coclass (group theory)|coclass]]. The other-maximal-cyclic group of order 2<sup>n</sup> always has nilpotency class 2. This makes the other-maximal-cyclic group less interesting, since most groups of order ''p''<sup>''n''</sup> for large ''n'' have nilpotency class 2 and have proven difficult to understand directly. <!-- I need to go get a copy of this folklore statement. the result should be in {{MR|113948}} or {{MR|169921}} -->
Through the moment you begin playing Amea, you might think the overall game is often a happy sort of game. Needless to say, therefore, basketball players during the period of infancy of basketball as a sport only have a couple of jerseys and the team owners expect that the jerseys last within the whole year. They include stickman sniper games and the stickman shooting games. The story and presentation in Call of Duty: [http://www.google.de/search?q=Black+Ops Black Ops] is fairly well done, though pales in comparison to those presented in the Modern Warfare variant of the Call of Duty series.


The generalized quaternion, the dihedral, and the quasidihedral group are the only 2-groups whose [[derived subgroup]] has index 4. The [[Alperin–Brauer–Gorenstein theorem]] classifies the simple groups, and to a degree the finite groups, with quasidihedral Sylow 2-subgroups.
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==References==
*{{cite book
| last1 = Dummit
| first1 = D. S.
| last2 = Foote
| first2 = R.
| title = Abstract Algebra
| publisher = Wiley
| pages = 71-72
| year = 2004
| edition = 3
| isbn = 9780471433347
}}
* {{cite book
| last = Huppert
| first = B.
| authorlink = Bertram Huppert
| title = Endliche Gruppen
| id = {{MathSciNet | id = 0224703}}
| year = 1967
| publisher = Springer
| pages = 90–93
}}
* {{cite book
| last = Gorenstein
| first = D.
| authorlink = Daniel Gorenstein
| title = Finite Groups
| id = {{MathSciNet | id = 81b:20002}}
| year = 1980
| publisher = Chelsea
| isbn = 0-8284-0301-5
| pages = 188–195
}}
 
[[Category:Finite groups]]

Latest revision as of 20:55, 12 January 2015


I soon realized the complexities in not only understanding these external and internal forces in a sound golf swing, but also in the managing of these forces. You may be partially right when you scold your child to stop playing action games in the computer throughout the day because that may have an adverse effect on their health, just like any other thing in this world, action games also have a good and a bad side. The low roller must place 5 coins on either row 1 or row 7, with one being in each of the squares; the high roller places two coins on the same row. 

This third and first-person shooter allows players to take on the role of Jaden Korr, a student of Kyle Katarn's at the Jedi Academy. For some it may be all they'll ever play, but for most this will be just a starting point. You can purchase prepared bagel pizzas, wings, and chicken nuggets to throw in the over and then have them ready when the guests arrive. At some time, women perform just like men in this skill, mostly applied in mathematics and engineering subjects.

Some of these games that come under adventurous or action games have created hallmark of such genre. We're currently working on a large milestone release, which will include new worlds, and up to four-player multiplayer versus mode. Most people who own the cheap imitation guns are often in a spot of bother when they find that the gun slows down on performance in some time. Games are playing since that time technology doesn't advantages a lot that people notice nowadays.

Kizi 2 games have also a vast package of entertainment, which also increases your skills of motoring, skills of reasoning, analytical skills, and other skills of language. The prime factors of originality for Gunz falls within the ability to run up walls; run sideways on walls; fight in mid-air with swords and dual-wield a number of single-handed guns and blades. Reading the table is fairly simple once you know how.

Through the moment you begin playing Amea, you might think the overall game is often a happy sort of game. Needless to say, therefore, basketball players during the period of infancy of basketball as a sport only have a couple of jerseys and the team owners expect that the jerseys last within the whole year. They include stickman sniper games and the stickman shooting games. The story and presentation in Call of Duty: Black Ops is fairly well done, though pales in comparison to those presented in the Modern Warfare variant of the Call of Duty series.

There are several modes that you could enjoy including single player mode, coop mode, and the best of all, multi player wars. Description: Futuristic shooter with multiple classes and more than 40 weapons. Anything that makes workouts more varied helps keep boredom beat back. You can get certain benefits and helps if you just visit your neighbors on a daily basis. They performed better in tests involving speed, multitasking and others. When we talk about flash-based versions, we are basically talking about games that have been created utilizing a uniform standard that was developed by Adobe Systems Incorporated.

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