現代数学の系譜工学物理雑談古典ガロア理論も読む78

[過去ﾛｸﾞ] 現代数学の系譜工学物理雑談古典ガロア理論も読む78 (1002ﾚｽ)
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18: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [sage] 2019/10/19(土) 19:25:01.03 ID:ti2BclkQ

S5の位数20の部分群
https://groupprops.subwiki.org/wiki/General_affine_group:GA(1,5)
General affine group:GA(1,5)
(抜粋)
As GA(1,q), q = 5: q(q - 1) = 5(5 - 1) = 20
As holomorph of cyclic group:Z5: |Z5||Aut(Z5)| = 5・4 = 20
As Sz(q), q = 2: q^2(q^2 + 1)(q - 1) = 2^2(2^2 + 1)(2 - 1) = 4・5・1 = 20

Group properties
Function Value
abelian group No
nilpotent group No
metacyclic group Yes
supersolvable group Yes
solvable group Yes
Frobenius group Yes
Camina group Yes

https://people.maths.bris.ac.uk/~matyd/
Tim Dokchitser Arithmetic/Algebraic Geometry University of Bristol
https://people.maths.bris.ac.uk/~matyd/GroupNames/1/F5.html
G = F5? order 20 = 2^2・5 Frobenius group Tim Dokchitser

https://groupprops.subwiki.org/wiki/General_affine_group_of_degree_one
General affine group of degree one
GA(1,K) = K semix K^*
https://ja.wikipedia.org/wiki/%E3%82%A2%E3%83%95%E3%82%A3%E3%83%B3%E7%BE%A4
アフィン群

https://en.wikipedia.org/wiki/Frobenius_group
Frobenius group
(抜粋)
In mathematics, a Frobenius group is a transitive permutation group on a finite set, such that no non-trivial element fixes more than one point and some non-trivial element fixes a point. They are named after F. G. Frobenius.

Structure
A 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 (1901); there is still no proof of this theorem that does not use character theory, although see [1].)
The Frobenius group G is the semidirect product of K and H:
G=K semix H

つづく

http://rio2016.5ch.net/test/read.cgi/math/1571400076/18

19: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [sage] 2019/10/19(土) 19:25:28.89 ID:ti2BclkQ

>>18

つづき

(スレ77 https://rio2016.5ch.net/test/read.cgi/math/1568026331/884- より)
http://www.isc.meiji.ac.jp/~kurano/soturon/ronbun/08kurano.pdf
2008 年度卒業研究 S_3, S_4, S_5 の部分群の分類
(抜粋)
P3
S5の部分群
位数20： < (12345), (2354) > S5中の共役な群6個

http://www.isc.meiji.ac.jp/~kurano/soturon/ronbun/04kurano.pdf
2004 年度卒業研究 位数 30 以下の群の分類
(抜粋)
P3
位数20 5個；
アーベル：C4 × C5, C2 × C2 × C5 ２個,
非アーベル： C5 semix C4, Q20, D20 ３個
P16
11 位数 20 の群の分類

https://en.wikipedia.org/wiki/List_of_small_groups
List of small groups
(抜粋)
List of small abelian groups
位数20
51 G202 Z20 = Z5 × Z4 Z10, Z5, Z4, Z2 GroupDiagramMiniC20.svg Cyclic. Product.
54 G205 Z10 × Z2 = Z5 × Z22 Z5, Z2 GroupDiagramMiniC2C10.png Product.

List of small non-abelian groups
位数20
50 G201 Q20 = Dic5 = <5,2,2> GroupDiagramMiniQ20.png Binary dihedral group
52 G203 Z5 semix Z4 GroupDiagramMiniC5semiprodC4.png Frobenius group
53 G204 Dih10 = Dih5 × Z2 = D20 GroupDiagramMiniD20.png Dihedral group, product
(引用終り)
以上

http://rio2016.5ch.net/test/read.cgi/math/1571400076/19

58: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [] 2019/10/20(日) 13:28:47.86 ID:f+LcfVi/

>>18
Terence TaoのFrobenius group追加

https://terrytao.wordpress.com/2013/04/12/the-theorems-of-frobenius-and-suzuki-on-finite-groups/
What's new
Updates on my research and expository papers, discussion of open problems, and other maths-related topics. By Terence Tao
Tag Archive
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The theorems of Frobenius and Suzuki on finite groups
12 April, 2013 in expository, math.GR, math.RT | Tags: CA groups, characters, classification of finite simple groups, Fourier transform, Frobenius groups, Frobenius theorem, induced representations, integrality gap, Suzuki theorem
略

3 June, 2013 at 1:03 pm
Terence Tao
Yes, this is something I would like to understand better myself.
One of the funny things coming out of Suzuki’s analysis is that to every (Weyl group conjugacy class of a) character \xi_{i,a} on a (conjugacy class of a) maximal abelian subgroup H_i of G there is associated an “exceptional character” \xi^*_{i,a} of G which is a component of the induced representation of G coming from the character of H_i (or sometimes, a bit weirdly,
it is an “anti-component”, if the sign \epsilon_i is negative), and略

27 June, 2013 at 12:19 pm
Terence Tao
The original reference is
G. Frobenius, “Ueber auflosbare Gruppen IV” Sitzungsber. Preuss. Akad. Wissenschaft. (1901) pp. 1216?1230
but it may be difficult to locate (see http://math.stackexchange.com/questions/222167/where-can-i-find-the-original-papers-by-frobenius-concerning-solutions-to-xn for some related discussion). A somewhat more modern reference is
I.M. Isaacs, “Character theory of finite groups” , Acad. Press (1976)

http://rio2016.5ch.net/test/read.cgi/math/1571400076/58

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