作者xcycl (XOO)
看板Math
标题Re: [代数] direct factor
时间Wed Mar 16 19:49:01 2011
※ 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.
: (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.
: 以上为Hungerford代数的习题
: 恳请板上高手协助 感谢!!!!
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