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一道抽象代数教科书上的难题

2023-08-02 06:34 作者:南海之声sonnet耳放  | 我要投稿

G%3D%3Cg_1%2Cg_2%2C...%2Cg_n%3E%2C%20%5BG%3AA%5D%3Dm%2CA%E5%8F%AF%E4%BB%A5%E7%94%B12nm%E4%B8%AA%E5%85%83%E7%B4%A0%E7%94%9F%E6%88%90

考虑陪集分解:A%5Ccup%20A_%7Ba_2%7D%5Ccup%20...%5Ccup%20A_%7Ba_m%7D,任意元素都可以表示成g_1%2Cg_2%2C...%2Cg_n%2Cg_1%5E%7B-1%7D%2Cg_2%5E%7B-1%7D%2C...g_n%5E%7B-1%7D的乘积组合,干脆定义g_%7Bn%2Bi%7D%3Dg_i%5E%7B-1%7D将指标扩展到2n

定义一些A中的元素b_%7Bij%7D去满足等式a_ig_j%3Db_%7Bij%7Da_k%2Ci%3D1%2C2%2C...%2Cn%2Ck%3D1%2C2%2C...%2Cm%2Cj%3D1%2C2%2C...%2C2n

A%5E%5Cprime%3D%3Cb_%7Bik%7D%3E%2C%E8%AF%95%E5%9B%BE%E8%AF%81%E6%98%8E%EF%BC%9AG%3DA%5E%5Cprime%20%5Ccup%20A%5E%5Cprime%7Ba_2%7D%5Ccup%20...%5Ccup%20A%5E%5Cprime%7Ba_m%7D%20%E5%8D%B3%E5%8F%AF%E8%AF%81%E6%98%8EA%3DA%5E%5Cprime,

只需证明任何G元素都可以表示为形如某个ba_k,而G的任何元素都是g_i的乘积组合,

gggg%3Dbaggg,只需证明aggg%3Dba,这是显然的。


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