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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30: 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) 22:02:00.68 ID:ti2BclkQ

メモ

https://en.wikipedia.org/wiki/Projective_linear_group
Projective linear group
(抜粋)
PGL(V) = GL(V)/Z(V)
where GL(V) is the general linear group of V and Z(V) is the subgroup of all nonzero scalar transformations of V

PSL(V) = SL(V)/SZ(V)
where SL(V) is the special linear group over V and SZ(V) is the subgroup of scalar transformations with unit determinant.

PGL and PSL are some of the fundamental groups of study, part of the so-called classical groups, and an element of PGL is called projective linear transformation, projective transformation or homography.
If V is the n-dimensional vector space over a field F, namely V = Fn, the alternate notations PGL(n, F) and PSL(n, F) are also used.

there are other exceptional isomorphisms between projective special linear groups and alternating groups (these groups are all simple, as the alternating group over 5 or more letters is simple):

L_2(4) ＝～ A_5
L_2(5) ＝～ A_5 (see here for a proof)

つづく

http://rio2016.5ch.net/test/read.cgi/math/1571400076/30

31: 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) 22:02:50.25 ID:ti2BclkQ

>>30

つづき

The groups over F5 have a number of exceptional isomorphisms:

PSL(2, 5) ＝～ A5 ＝～ I, the alternating group on five elements, or equivalently the icosahedral group;
PGL(2, 5) ＝～ S5, the symmetric group on five elements;
SL(2, 5) ＝～ 2 ・ A5 ＝～ 2I the double cover of the alternating group A5, or equivalently the binary icosahedral group.

They can also be used to give a construction of an exotic map S5 → S6, as described below. Note however that GL(2, 5) is not a double cover of S5, but is rather a 4-fold cover.

・PSL(2, 4) = PGL(2, 4) → S5, of order 60, yielding the alternating group A5.
・PSL(2, 5) < PGL(2, 5) → S6, of orders 60 and 120, which yields an embedding of S5 (respectively, A5) as a transitive subgroup of S6 (respectively, A6).
This is an example of an exotic map S5 → S6, and can be used to construct the exceptional outer automorphism of S6.[6]
Note that the isomorphism PGL(2, 5) ＝～ S5 is not transparent from this presentation: there is no particularly natural set of 5 elements on which PGL(2, 5) acts.

・L_2(5) ＝～ A_5. To construct such an isomorphism, one needs to consider the group L2(5) as a Galois group of a Galois cover a5: X(5) → X(1) = P1,
(引用終り)
以上

http://rio2016.5ch.net/test/read.cgi/math/1571400076/31

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