※ 引述《hotplushot (热加热)》之铭言： : 先述说一个定理 : 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 建造一个abelian group F使其有basis,自然F就是自由群 : 第二步 证明G为F的子群 根据上述定理 自然G就是自由交换群 : 请问这想法对吗? : 如果对 要怎麽更确切写出来(如果对 我觉得第一步比较难写出来) : 2. : The direct sum of a family of free abelian group is free abelian. : 这题暂时没什麽头绪 : 请版友能给予协助 感激不尽 2. 我不知道这样写能不能理解 ... 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. 唯一你要检验的是，Abelian group 的 coproduct 跟 group 的 direct sum 是一样的。 -- Welcome to the dark side. --

※ 发信站: 批踢踢实业坊(ptt.cc) ◆ From: 78.109.182.40 ※ 编辑: xcycl 来自: 78.109.182.40 (03/07 07:24)