Re: [代數] Hungerford代數上的習題
※ 引述《hotplushot (熱加熱)》之銘言：
: 先述說一個定理
: Thm
: If F is a free abelian group of finite rank n and G is a nonzero subgroup of F
: then there exists a basis {x(1),...,x(n)} of F,an integer r(1≦r≦n)
: and positive integers d(1),...,d(r) such that d(1)|d(2)|...|d(r)
: and G is free abelian with basis{d(1)x(1),...,d(r)x(r)}.
: 1.
: Let G be a finitely generated abelian group in which no element(except 0)
: has finite order.
: Then G is a free abelian group.(提示:使用上面定理)
: 我的想法:
: 第一步 建造一個abelian group F使其有basis,自然F就是自由群
: 第二步 證明G為F的子群 根據上述定理 自然G就是自由交換群
: 請問這想法對嗎?
: 如果對 要怎麼更確切寫出來(如果對 我覺得第一步比較難寫出來)
YES. Let X={x_1,...,x_n} be a set of generators of G.
Consider F(X) and the natural map f:F(X)-> G. Apply the theorem
to ker f. Here F(X) is the free abelian group generated by X
and the map f is the epimorphism given by x_i->x_i.
: 2.
: The direct sum of a family of free abelian group is free abelian.
: 這題暫時沒什麼頭緒
: 請版友能給予協助 感激不盡
Can you find a basis for the direct sum? if you have a basis for
each member of the family. This is my idea but you can have different idea.
我的基礎代數忘的差不多了~~所以僅供參考~~~XD
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◆ From: 128.120.178.219
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03/07 10:30, , 1F
03/07 10:30, 1F
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03/07 10:30, , 2F
03/07 10:30, 2F
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03/07 10:41, , 3F
03/07 10:41, 3F
推
03/07 15:25, , 4F
03/07 15:25, 4F
※ 編輯: herstein 來自: 128.120.178.219 (03/07 16:35)
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03/07 17:22, , 5F
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03/07 17:39, , 9F
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03/07 17:50, , 10F
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03/07 17:51, 11F
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03/07 17:53, , 13F
03/07 17:53, 13F
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