Cynosure对《Geometries》的笔记(1)

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个人觉得用几何来学群论比较通俗，比较三本讲几何和群论的书：
https://book.douban.com/subject/30322367/，群论占大比重
https://book.douban.com/subject/20277843/，群论和几何的分量相当
https://book.douban.com/subject/10073323/，几何占大比重。是text book。每种几何讲得很细致，比如有很多应用的例题。
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Nice mathematical objects often have naturally defined “quotient objects” obtained by “dividing out” the given object by some subobject。。。The construction of “quotient groups” along those lines works only when the subgroup used is in a sense “nice”,
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Affine spaces are 。。。 finite-dimensional vector spaces “without a fixed origin”.
Projective spaces RP^n are obtained from affine spaces by“adding points at infinity” 
This means that any contradiction resulting from Hilbert’s systemof axioms must also appear in the arithmetic related to Ω. Thus wecan say that Hilbert’s axiom system is consistent provided that thereare no contradictions in the theory of algebraic numbers.
duality principle, dual theorem，  Projective duality
12.5.3. Corollary: Duality Principle. There is a bijection between the set of lines and the set of points of RP 2 that preservesincidence and takes any theorem of projective geometry to a theoremof projective geometry.