※ 引述《hotplushot (熱加熱)》之銘言：
: http://tinyurl.com/4vns4gz
: 裡面的第一題
: 很難去感覺子群跟Zp⊕Zp同構
: 不太知道怎麼去寫證明
: 煩請版友高手給個方向
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.
僅供參考~
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