作者VFresh (车干)
看板Math
标题Re: [代数] 习题烦请给个方向
时间Wed Mar 30 03:21:57 2011
※ 引述《hotplushot (热加热)》之铭言：
: http://tinyurl.com/4vns4gz
: 里面的第一题
: 很难去感觉子群跟Zp⊕Zp同构
: 不太知道怎麽去写证明
: 烦请版友高手给个方向
Let G be a finite abelian group and G is NOT cyclic.
Then G is isomorphic to Zn_1⊕Zn_2⊕...⊕Zn_k, where n_1|n_2|...|n_k.
We may let f be the isomorphism from G onto Zn_1⊕...⊕Zn_k
If k = 1, then G is cyclic; hence, k ≧2.
We can find a prime p such that p|n_1.
Consider p|n_1|n_2, then p|n_2.
Let s_1 = n_1/p, s_2 = n_2/p.
Let H = (s_1)Zn_1⊕(s_2)Zn_2⊕(0)⊕...⊕(0).
Then f^-1(H) is a subgroup which is isomorphic to Zp⊕Zp.
仅供参考~
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