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)
上下前次1-新
通常表示 512ﾊﾞｲﾄ分割ﾚｽ栞
抽出解除ﾚｽ栞

このｽﾚｯﾄﾞは過去ﾛｸﾞ倉庫に格納されています｡
次ｽﾚ検索歴削→次ｽﾚ栞削→次ｽﾚ過去ﾛｸﾞﾒﾆｭｰ

379: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [] 2019/11/02(土) 10:35:48.60 ID:ZLEqKHqI

>>378

つづき

The action of L2(11) can be seen algebraically as due to an exceptional inclusion L2(5)→ L2(11) - there are two conjugacy classes of subgroups of L2(11) that are isomorphic to L2(5),
each with 11 elements: the action of L2(11) by conjugation on these is an action on 11 points,
and, further, the two conjugacy classes are related by an outer automorphism of L2(11). (The same is true for subgroups of L2(7) isomorphic to S4, and this also has a biplane geometry.)

Geometrically, this action can be understood via a biplane geometry, which is defined as follows.
A biplane geometry is a symmetric design (a set of points and an equal number of "lines", or rather blocks) such that any set of two points is contained in two lines, while any two lines intersect in two points;
this is similar to a finite projective plane, except that rather than two points determining one line (and two lines determining one point),
they determine two lines (respectively, points). In this case (the Paley biplane, obtained from the Paley digraph of order 11),
the points are the affine line (the finite field) F11,
where the first line is defined to be the five non-zero quadratic residues (points which are squares: 1, 3, 4, 5, 9),
and the other lines are the affine translates of this (add a constant to all the points).
L2(11) is then isomorphic to the subgroup of S11 that preserve this geometry (sends lines to lines), giving a set of 11 points on which it acts - in fact two:
the points or the lines, which corresponds to the outer automorphism - while L2(5) is the stabilizer of a given line, or dually of a given point.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1571400076/379

380: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [] 2019/11/02(土) 10:36:15.87 ID:ZLEqKHqI

>>379

つづき

More surprisingly, the coset space L2(11)/Z/11Z, which has order 660/11 = 60 (and on which the icosahedral group acts) naturally has the structure of a buckeyball, which is used in the construction of the buckyball surface.

History
The groups PSL(2, p) were constructed by Evariste Galois in the 1830s,
and were the second family of finite simple groups, after the alternating groups.[3]
Galois constructed them as fractional linear transforms, and observed that they were simple except if p was 2 or 3;
this is contained in his last letter to Chevalier.[4]
In the same letter and attached manuscripts, Galois also constructed the general linear group over a prime field, GL(ν, p), in studying the Galois group of the general equation of degree p^ν.
The groups PSL(n, q) (general n, general finite field) were then constructed in the classic 1870 text by Camille Jordan, Traite des substitutions et des equations algebriques.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1571400076/380

上下前次1-新書関写板覧索設栞歴

ｽﾚ情報赤ﾚｽ抽出画像ﾚｽ抽出歴の未読ｽﾚ AAｻﾑﾈｲﾙ

ぬこの手ぬこTOP 2.516s*