作者hotplushot (热加热)
看板Math
标题Re: [代数] direct factor
时间Wed Mar 16 20:41:19 2011
※ 引述《xcycl (XOO)》之铭言：
: ※ 引述《hotplushot (热加热)》之铭言：
: : Def:
: : A normal subgroup H of a group G is said to be a direct factor
: : if there exists a (normal)subgroup K of G such that G=H×K
: : problem:
: : (a)If H is a direct factor of K and K is a direct factor of G,
: : then H is normal in G.
: K = H x K'
: G = K x G'
: Therefore, G = (H x K') x G' which is isomorphic to
: G = H x (K' x G'), so H is a direct factor of G and normal in G.
几个小问题想请教
1.(H x K') x G'= H x (K' x G')的原因是??
後来想想 这好像很直观 是因为群本身就有结合律吧......
经xcycl大大指正 是根据universal property
不是结合律的关系
2.从H x (K' x G')则推到 H normal in G 我不太能理解
因为G = H x (K' x G')只说明 G 可以是 H与(K' x G')的direct product
: : (b)If H is a direct factor of G,then every homomorphism H → G may be
: : extended to an endomorphism G → G.
: Hint: H → G is equivalent to say H → H x K.
: : However,a monomorphism H → G need not be extendible to an automorphism
: : G → G.
根据大大提示 我证明看看
http://tinyurl.com/4drgp7g
: : 以上为Hungerford代数的习题
: : 恳请板上高手协助 感谢!!!!
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◆ From: 111.252.196.205
※ 编辑: hotplushot 来自: 111.252.196.205 (03/16 20:45)
※ 编辑: hotplushot 来自: 111.252.196.205 (03/16 20:47)
※ 编辑: hotplushot 来自: 111.252.196.205 (03/16 21:42)
※ 编辑: hotplushot 来自: 111.252.196.205 (03/16 22:42)
1F：推 recorriendo :2. direct product两个factor都会normal 这是direct 03/17 01:44
2F：→ recorriendo :product的重要性直 03/17 01:44
internal direct product才可以这样说吧....
※ 编辑: hotplushot 来自: 111.252.210.30 (03/17 18:47)
3F：→ xcycl :一般都会讲 up to isomorphism 来避免区分这类问题 03/18 04:59
4F：→ xcycl :internal/external 分别意义不大 03/18 05:01
5F：→ xcycl :但谈 subgroup 就满微妙的, 是要定义作子集才算 03/18 05:09
6F：→ xcycl :还是同构的都算，前者好理解，後者则是方便 03/18 05:10
7F：→ xcycl :up to isomorphism 下, 可以说 N is normal in NxH 03/18 05:35