Re: [代數] direct factor
※ 引述《xcycl (XOO)》之銘言：
: ※ 引述《hotplushot (熱加熱)》之銘言：
: : 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.
幾個小問題想請教
1.(H x K') x G'= H x (K' x G')的原因是??
後來想想 這好像很直觀 是因為群本身就有結合律吧......
經xcycl大大指正 是根據universal property
不是結合律的關係
2.從H x (K' x G')則推到 H normal in G 我不太能理解
因為G = H x (K' x G')只說明 G 可以是 H與(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.
根據大大提示 我證明看看
http://tinyurl.com/4drgp7g
: : 以上為Hungerford代數的習題
: : 懇請板上高手協助 感謝!!!!
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◆ From: 111.252.196.205
※ 編輯: hotplushot 來自: 111.252.196.205 (03/16 20:45)
※ 編輯: hotplushot 來自: 111.252.196.205 (03/16 20:47)
※ 編輯: hotplushot 來自: 111.252.196.205 (03/16 21:42)
※ 編輯: hotplushot 來自: 111.252.196.205 (03/16 22:42)
推
03/17 01:44, , 1F
03/17 01:44, 1F
→
03/17 01:44, , 2F
03/17 01:44, 2F
internal direct product才可以這樣說吧....
※ 編輯: hotplushot 來自: 111.252.210.30 (03/17 18:47)
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03/18 04:59, , 3F
03/18 04:59, 3F
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03/18 05:09, , 5F
03/18 05:09, 5F
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03/18 05:35, , 7F
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