作者VFresh (车干)
看板Math
标题Re: [代数] 自由群的问题
时间Fri Mar 18 02:16:48 2011
方法大同小异 仅供参考-
Let F be a free abelian group.
We can find a basis {e_i}, i is in an index set I.
Then F = Σ <e_i>.
i in I
Let j be in I.
Let A = Σ Ci, Ci = <e_i> if i≠j, Cj = <n*e_j>.
i in I
(Where n*e_j = e_j +...+ e_j (n-times))
Clearly, A is a subgroup of F and
F/A is isomorphic to Σ Si, Si = <e_i>/<e_i> if i≠j, Sj = <e_j>/<n*e_j>
i in I
Then F/A is isomorphic to Zn since <e_j>/<n*e_j> is isomorphic to Zn,
hence, [F:A] = n.
※ 引述《hotplushot (热加热)》之铭言：
: ※ 引述《hotplushot (热加热)》之铭言：
: : A nonzero free abelian group has a subgroup of index n
: : for all positive integer n.
: : 这题毫无头绪.....
: : 可否请版友协助
: : 感谢!!!!
: because F is free abelian
: F=Z⊕...⊕Z(m个)
: let A=Z⊕...⊕nZ⊕...⊕Z
: [F:A]=n
: and nZ is subgroup of Z
: so conclusion is hold.
: 是这样吗
--
※ 发信站: 批踢踢实业坊(ptt.cc)
◆ From: 111.251.166.99
1F：推 sato186 :啊 被抢PO了 03/18 02:51
2F：→ VFresh :so? 03/18 05:59
3F：→ VFresh :你高兴可以再PO一篇 我这篇砍掉 03/18 06:00