※ 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)
: 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
2. 我不知道這樣寫能不能理解 ...
Consider the adjunction (F, G, \phi) between the category Set of sets and
the category Ab of abelian groups where FX is the free abelian group of a
set X and G is the forgetful functor sending a group to its carrier set.
Since F is a left adjoint, it preserves colimits.
By applying F to the disjoint union X of X_i, i.e. the coproduct of X_i in Set,
we get F(X) which is the free abelian group of X and is isomorphic
to the coproduct of F(X_i) in Ab.
唯一你要檢驗的是，Abelian group 的 coproduct 跟 group 的 direct sum 是一樣的。
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◆ From: 78.109.182.40
※ 編輯: xcycl 來自: 78.109.182.40 (03/07 07:24)