[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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378(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)10:34 ID:ZLEqKHqI(9/18) AAS
>>377

つづき

　One then needs to consider the action of the symmetry group of icosahedron on the five associated tetrahedra.

・L2(7) =~ L3(2) which acts on the 1+2+4 = 7 points of the Fano plane (projective plane over F2); this can also be seen as the action on order 2 biplane, which is the complementary Fano plane.
・L2(11) is subtler, and elaborated below; it acts on the order 3 biplane.[8]
Further, L2(7) and L2(11) have two inequivalent actions on p points; geometrically this is realized by the action on a biplane,
which has p points and p blocks - the action on the points and the action on the blocks are both actions on p points,
省7

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379(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)10:35 ID:ZLEqKHqI(10/18) AAS
>>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;
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