[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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376(3): 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)10:30 ID:ZLEqKHqI(7/18) AAS
>>375
つづき
Action on projective line
Some of the above maps can be seen directly in terms of the action of PSL and PGL on the associated projective line:
PGL(n, q) acts on the projective space Pn?1(q), which has (q^n?1)/(q?1) points,
and this yields a map from the projective linear group to the symmetric group on (q^n?1)/(q?1) points.
For n = 2, this is the projective line P1(q) which has (q^2?1)/(q?1) = q+1 points,
省13
377(1): 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)10:33 ID:ZLEqKHqI(8/18) AAS
>>376
つづき
・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.
Action on p points
While PSL(n, q) naturally acts on (qn?1)/(q?1) = 1+q+...+qn?1 points, non-trivial actions on fewer points are rarer. I
ndeed, for PSL(2, p) acts non-trivially on p points if and only if p = 2, 3, 5, 7, or 11; for 2 and 3 the group is not simple, while for 5, 7, and 11, the group is simple - further, it does not act non-trivially on fewer than p points.[note 5] This was first observed by Evariste Galois in his last letter to Chevalier, 1832.[7]
省8
411: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/11/03(日)08:52 ID:apiWSBWV(12/33) AAS
>>409
式はわかるよ >>375-376にあるでしょ？
4080 からの逆算がね（＾＾
422(1): 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/11/03(日)09:16 ID:apiWSBWV(20/33) AAS
>>420
＞スレ主が分からないのは本当だろうけど
別に
全部 >>375-376にある
それと、>>396の説明もあるよ
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