Coset & Index 陪集,指数

群论的原本是Galois 发明,没有现在这么抽象的Axiom-based, 是100年后 Noether 在 WW2 归纳的抽象化。Galois 只发明 左/右陪集 Left /Right Coset (l’ensemble à gauche / à droite),当 左陪集=右陪集,就是Normal subgroup (正规子群,l’invariant), 破解300年的数学难题:5次方程以上(Quintic equations & above ) 无根式 (radical root) 解, 从而诞生Group Theory, 开辟抽象代数Abstract Algebra / New Math.

这位中国年轻老师 教得很棒, 证明严谨rigorous, 很好的方法:eg. Test “well-defined”, necessary & sufficient condition (Set Prove Technique : ⊂ left inclusion, ⊃ right inclusion, then equal =), Bijection 双射 (Surjective 满射 trivial + Injective proof).

|Z : nZ|Index is easy ?
把 Z 用coset 分类 (partitioned) 为
0Z,1Z, 2Z,… (n-1)Z 个陪集,
Total = n个

|Z : nZ|= n

Note: Coset in Calculus ( Indefinite Integration) :

Integral Solution “+ C” 就是 Right Coset = solution的右陪集:意思所有 “+C” 都是solution Set 的 右陪伴。

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