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| In [[mathematics]], a '''Frobenius group''' is a [[group action#Types of actions|transitive]] [[permutation group]] on a [[finite set]], such that no non-trivial element
| | == our plan in accordance with this hang.' == |
| fixes more than one point and some non-trivial element fixes a point.
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| They are named after [[Ferdinand Georg Frobenius|F. G. Frobenius]].
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| == Structure ==
| | A swirl channel [http://www.nrcil.net/fancybox/lib/rakuten_LV_117.html ルイヴィトン旅行バッグ] leading to the other side of that.<br><br>approximation with a streamer.<br><br>'Venerable Man nest came with his field class treasure ...... we also have a third re-arranged, all the more confident. despite sky monster that another powerful tool, even Guards special life, we six Even if no match cosmic overlord, by virtue of [http://www.nrcil.net/fancybox/lib/rakuten_LV_15.html ルイヴィトン コレクション] the many means absolutely can trap him and he absolutely can not escape! '<br><br>'the news came on the sinking of the [http://www.nrcil.net/fancybox/lib/rakuten_LV_75.html ルイヴィトンのバッグ] original star, hung [http://www.nrcil.net/fancybox/lib/rakuten_LV_93.html ルイヴィトンレディース財布] some of the strong alliance seems to [http://www.nrcil.net/fancybox/lib/rakuten_LV_50.html ルイヴィトン ジッピーオーガナイザー] be ready to kill the two stars monster.'<br><br>'trouble.'<br><br>'how do we do in this [http://www.nrcil.net/fancybox/lib/rakuten_LV_74.html ルイヴィトンの財布] hang?'<br><br>'assured that the original Star big enough, they can not teleport, transfer the kingdom of God, which is so good Weisha of? our plan in accordance with this hang.'<br><br>......<br><br>swirl passage [http://www.nrcil.net/fancybox/lib/rakuten_LV_126.html ルイヴィトン 値段] to the outside world, in the next Fangshan Lin.<br><br>'Qi blood Miyaji, Peel Feng Venerable, the two of [http://www.nrcil.net/fancybox/lib/rakuten_LV_109.html 財布 ルイヴィトン メンズ] you came, arguably the fastest to reach you |
| The [[subgroup]] ''H'' of a Frobenius group ''G'' fixing a point of the set ''X'' is called the '''Frobenius complement'''. The identity element together with all elements not in any conjugate of ''H'' form a [[normal subgroup]] called the '''Frobenius kernel''' ''K''. (This is a theorem due to Frobenius; there is still no proof of this theorem that does not use [[character theory]].) The Frobenius group ''G'' is the [[semidirect product]] of ''K'' and ''H'':
| | 相关的主题文章： |
| :<math>G=K\rtimes H</math>.
| | <ul> |
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| | <li>[http://www.xuhui365.com/thread-1171167-1-1.html http://www.xuhui365.com/thread-1171167-1-1.html]</li> |
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| | <li>[http://www.dongshansi.org/plus/feedback.php?aid=293 http://www.dongshansi.org/plus/feedback.php?aid=293]</li> |
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| | <li>[http://eiga.yokkaichi.org/sunbbs2/sunbbs.cgi http://eiga.yokkaichi.org/sunbbs2/sunbbs.cgi]</li> |
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| | </ul> |
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| Both the Frobenius kernel and the Frobenius complement have very restricted structures. {{harvs|txt|authorlink=John_Griggs_Thompson|first=J. G.|last=Thompson|year=1960}} proved that the Frobenius kernel ''K'' is a [[nilpotent group]]. If ''H'' has even order then ''K'' is abelian. The Frobenius complement ''H'' has the property that every subgroup whose order is the product of 2 primes is cyclic; this implies that its [[Sylow subgroup]]s are [[cyclic group|cyclic]] or [[quaternion group|generalized quaternion]] groups. Any group such that all Sylow subgroups are cyclic is called a [[Z-group#Groups whose Sylow subgroups are cyclic|Z-group]], and in particular must be a [[metacyclic group]]: this means it is the extension of two cyclic groups. If a Frobenius complement ''H'' is not solvable then [[Hans Zassenhaus|Zassenhaus]] showed that it
| | == we have entered the beginning Fam == |
| has a normal subgroup of [[Index of a subgroup|index]] 1 or 2 that is the product of SL<sub>2</sub>(5) and a metacyclic group of order coprime to 30. In particular, if a Frobenius complement coincides with its derived subgroup, then it is isomorphic with SL(2,5). If a Frobenius complement ''H'' is solvable then it has a normal metacyclic subgroup such that the quotient is a subgroup of the symmetric group on 4 points. A finite group is a Frobenius complement if and only if it has a faithful, finite-dimensional representation over a finite field in which non-identity group elements correspond to linear transformations without nonzero fixed points.<!-- Ito's Notes, Principal Theorem IV, p38 -->
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| The Frobenius kernel ''K'' is uniquely determined by ''G'' as it is the [[Fitting subgroup]], and the Frobenius complement is uniquely determined up to conjugacy by the [[Schur-Zassenhaus theorem]]. In particular a finite group ''G'' is a Frobenius group in at most one way.
| | Can not afford, do not worry you still first, serious practice, later to become my virtual universe, such as the company's super-strong, [http://www.nrcil.net/fancybox/lib/rakuten_LV_68.html ルイヴィトンの新作バッグ] the company may give you free directly with F-class ship spaceship.<br><br>following group of young people suddenly curl one's lip.<br><br>super strong? Spacecraft equipped with F grade?<br><br>can free [http://www.nrcil.net/fancybox/lib/rakuten_LV_10.html ルイヴィトン カバン メンズ] equipment, that have reached what extent the strength of the front of the group is just stellar little guys really too far away.<br><br>'regressed.' Luo Feng could not help herself.<br><br>'Hey, crazy.' beside [http://www.nrcil.net/fancybox/lib/rakuten_LV_95.html ルイヴィトン ミニバッグ] the pretty card held out his star juvenile Leuca head whispered, 'Do not listen to this [http://www.nrcil.net/fancybox/lib/rakuten_LV_133.html ルイヴィトン 財布 モノグラム] guy brag about, the world is not such a thing, certainly have wanted to pay! This is my father say! but I tell you, this is definitely a virtual universe company holy human [http://www.nrcil.net/fancybox/lib/rakuten_LV_27.html ルイヴィトン シューズ] population, countless treasures [http://www.nrcil.net/fancybox/lib/rakuten_LV_28.html ルイヴィトン財布ランキング] Arcane more than that, but unfortunately was not able to enter the original Uncharted, but fortunately, we have entered the beginning Fam, the benefits are very [http://www.nrcil.net/fancybox/lib/rakuten_LV_92.html バッグ ルイヴィトン] much you know [http://www.nrcil.net/fancybox/lib/rakuten_LV_117.html ルイヴィトン 価格] later the. ' |
| | | 相关的主题文章： |
| == Examples ==
| | <ul> |
| [[Image:Fano plane.svg|thumb|right|120px|The Fano plane]]
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| *The smallest example is the symmetric group on 3 points, with 6 elements. The Frobenius kernel ''K'' has order 3, and the complement ''H'' has order 2.
| | <li>[http://hi809.com/forum.php?mod=viewthread&tid=172748&fromuid=63811 http://hi809.com/forum.php?mod=viewthread&tid=172748&fromuid=63811]</li> |
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| *For every [[finite field]] ''F<sub>q</sub>'' with ''q'' (> 2) elements, the group of invertible [[affine transformation]]s <math> x \mapsto ax+b </math>, <math> a\ne 0 </math> acting naturally on ''F<sub>q</sub>'' is a Frobenius group. The preceding example corresponds to the case ''F<sub>3</sub>'', the field with three elements.
| | <li>[http://www.yy29600.com/forum.php?mod=viewthread&tid=231286 http://www.yy29600.com/forum.php?mod=viewthread&tid=231286]</li> |
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| *Another example is provided by the subgroup of order 21 of the [[collineation|collineation group]] of the [[Fano plane]] generated by a 3-fold symmetry σ fixing a point and a cyclic permutation τ of all 7 points, satisfying στ =τ²σ. Identifying ''F''<sub>8</sub>* with the Fano plane, σ can be taken to be the restriction of the [[Frobenius automorphism]] σ(''x'')=''x''² of ''F''<sub>8</sub> and τ to be multiplication by any element not in the [[Characteristic (algebra)#Case of fields|prime field]] ''F''<sub>2</sub> (i.e. a generator of the [[finite field#Applications|cyclic multiplicative group]] of ''F''<sub>8</sub>). This Frobenius group acts [[Group action#Types of actions|simply transitively]] on the 21 [[flag (geometry)|flag]]s in the Fano plane, i.e. lines with marked points.
| | <li>[http://forum.opsa.info/activity http://forum.opsa.info/activity]</li> |
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| *The [[dihedral group]] of order 2''n'' with ''n'' odd is a Frobenius group with complement of order 2. More generally if ''K'' is any abelian group of odd order and ''H'' has order 2 and acts on ''K'' by inversion, then the [[semidirect product]] ''K.H'' is a Frobenius group.
| | </ul> |
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| *Many further examples can be generated by the following constructions. If we replace the Frobenius complement of a Frobenius group by a non-trivial subgroup we get another Frobenius group. If we have two Frobenius groups ''K''<sub>1</sub>.''H'' and ''K''<sub>2</sub>.''H'' then (''K''<sub>1</sub> × ''K''<sub>2</sub>).''H'' is also a Frobenius group.
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| *If ''K'' is the non-abelian group of order 7<sup>3</sup> with exponent 7, and ''H'' is the cyclic group of order 3, then there is a Frobenius group ''G'' that is an extension ''K.H'' of ''H'' by ''K''. This gives an example of a Frobenius group with non-abelian kernel. This was the first example of Frobenius group with nonabelian kernel (it was constructed by Otto Schmidt).
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| *If ''H'' is the group ''SL''<sub>2</sub>(''F''<sub>5</sub>) of order 120, it acts fixed point freely on a 2-dimensional vector space ''K'' over the field with 11 elements. The extension ''K.H'' is the smallest example of a non-[[solvable group|solvable]] Frobenius group.
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| *The subgroup of a [[Zassenhaus group]] fixing a point is a Frobenius group.
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| *Frobenius groups whose Fitting subgroup has arbitrarily large nilpotency class were constructed by Ito: Let ''q'' be a prime power, ''d'' a positive integer, and ''p'' a prime divisor of ''q'' −1 with ''d'' ≤ ''p''. Fix some field ''F'' of order ''q'' and some element ''z'' of this field of order ''p''. The Frobenius complement ''H'' is the cyclic subgroup generated by the diagonal matrix whose ''i,i'''th entry is ''z''<sup>''i''</sup>. The Frobenius kernel ''K'' is the Sylow ''q''-subgroup of GL(''d'',''q'') consisting of upper triangular matrices with ones on the diagonal. The kernel ''K'' has nilpotency class ''d'' −1, and the semidirect product ''KH'' is a Frobenius group.
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| == Representation theory==
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| The irreducible complex representations of a Frobenius group ''G'' can be read off from those of ''H'' and ''K''. There are two types of [[irreducible representation]]s of ''G'':
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| *Any irreducible representation ''R'' of ''H'' gives an irreducible representation of ''G'' using the quotient map from ''G'' to ''H'' (that is, as a [[restricted representation]]). These give the irreducible representations of ''G'' with ''K'' in their kernel.
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| *If ''S'' is any ''non-trivial'' irreducible representation of ''K'', then the corresponding [[induced representation]] of ''G'' is also irreducible. These give the irreducible representations of ''G'' with ''K'' not in their kernel.
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| == Alternative definitions ==
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| There are a number of group theoretical properties which are interesting on their own right, but which happen to be equivalent to the group possessing a permutation representation that makes it a Frobenius group.
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| * ''G'' is a Frobenius group if and only if ''G'' has a proper, nonidentity subgroup ''H'' such that ''H'' ∩ ''H''<sup>''g''</sup> is the identity subgroup for every ''g'' ∈ ''G'' − ''H'', ''i.e.'' ''H'' is a [[malnormal subgroup]] of ''G''.
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| This definition is then generalized to the study of trivial intersection sets which allowed the results on Frobenius groups used in the classification of [[CA group]]s to be extended to the results on [[CN group]]s and finally the [[Odd_order_theorem#An outline of the proof|odd order theorem]].
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| Assuming that <math>G = K\rtimes H</math> is the [[semidirect product]] of the normal subgroup ''K'' and complement ''H'', then the following restrictions on [[centralizer]]s are equivalent to ''G'' being a Frobenius group with Frobenius complement ''H'':
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| * The [[centralizer]] C<sub>''G''</sub>(''k'') is a subgroup of K for every nonidentity ''k'' in ''K''.
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| * C<sub>''H''</sub>(''k'') = 1 for every nonidentity ''k'' in ''K''.
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| * C<sub>''G''</sub>(''h'') ≤ H for every nonidentity ''h'' in H.
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| ==References== | |
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| *B. Huppert, ''Endliche Gruppen I'', Springer 1967
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| *I. M. Isaacs, ''Character theory of finite groups'', AMS Chelsea 1976
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| *D. S. Passman, ''Permutation groups'', Benjamin 1968
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| *{{Citation | last1=Thompson | first1=John G. | author1-link=John G. Thompson | title=Normal p-complements for finite groups | doi=10.1007/BF01162958 | mr=0117289 | year=1960 | journal=[[Mathematische Zeitschrift]] | issn=0025-5874 | volume=72 | pages=332–354}}
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| [[Category:Permutation groups]] | |
| [[Category:Finite groups]]
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our plan in accordance with this hang.'
A swirl channel ルイヴィトン旅行バッグ leading to the other side of that.
approximation with a streamer.
'Venerable Man nest came with his field class treasure ...... we also have a third re-arranged, all the more confident. despite sky monster that another powerful tool, even Guards special life, we six Even if no match cosmic overlord, by virtue of ルイヴィトン コレクション the many means absolutely can trap him and he absolutely can not escape! '
'the news came on the sinking of the ルイヴィトンのバッグ original star, hung ルイヴィトンレディース財布 some of the strong alliance seems to ルイヴィトン ジッピーオーガナイザー be ready to kill the two stars monster.'
'trouble.'
'how do we do in this ルイヴィトンの財布 hang?'
'assured that the original Star big enough, they can not teleport, transfer the kingdom of God, which is so good Weisha of? our plan in accordance with this hang.'
......
swirl passage ルイヴィトン 値段 to the outside world, in the next Fangshan Lin.
'Qi blood Miyaji, Peel Feng Venerable, the two of 財布 ルイヴィトン メンズ you came, arguably the fastest to reach you
相关的主题文章：
we have entered the beginning Fam
Can not afford, do not worry you still first, serious practice, later to become my virtual universe, such as the company's super-strong, ルイヴィトンの新作バッグ the company may give you free directly with F-class ship spaceship.
following group of young people suddenly curl one's lip.
super strong? Spacecraft equipped with F grade?
can free ルイヴィトン カバン メンズ equipment, that have reached what extent the strength of the front of the group is just stellar little guys really too far away.
'regressed.' Luo Feng could not help herself.
'Hey, crazy.' beside ルイヴィトン ミニバッグ the pretty card held out his star juvenile Leuca head whispered, 'Do not listen to this ルイヴィトン 財布 モノグラム guy brag about, the world is not such a thing, certainly have wanted to pay! This is my father say! but I tell you, this is definitely a virtual universe company holy human ルイヴィトン シューズ population, countless treasures ルイヴィトン財布ランキング Arcane more than that, but unfortunately was not able to enter the original Uncharted, but fortunately, we have entered the beginning Fam, the benefits are very バッグ ルイヴィトン much you know ルイヴィトン 価格 later the. '
相关的主题文章：