计算机问题求解一论题3-18一群同态基本定理2017年3月13日
计算机问题求解 – 论题3-18 - 群同态基本定理 2017年3月13日
问题1：我们为什么定义这个函数是“同构”？iso-morphologyTwo groups (G,)and (H,o)are isomorphic if there exists a one-to-oneand onto map :G→H such that the group operation is preserved; that is,d(a.b)=d(a)oΦ(b)同构其实可在forallaandbin G.If GisisomorphictoH任何代数结构is called an isomorphism.(系统)上讨论
问题1：我们为什么定义这个函数是“同构” ？ iso-morphology 同构其实可在 任何代数结构 (系统)上讨论
问题2：从这个定理中，你能解释我们为什么研究“同构”吗？Theorem 9.1 Let :G-H be an isomorphism of two groups. Thenthefollowing statements are true.1.o-1:H-→Gisanisomorphism2. |G|= |H|.3.If G is abelian. then H is abelian4.IfGis cyclic, thenH is cyclic5.If Ghasasubgroupof ordern,thenHhasasubgroupof order n
问题2：从这个定理中，你能解释我们为 什么研究“同构”吗？
如何判断两个系统的同构？Theorem 9.2 All cyclic groups of infinite order are isomorphic to Z.PRooF.Let Gbe a cyclic group with infiniteorder and suppose that a is agenerator of G.Definea map o:Z-G by : n-an.Then观察o(m +n) = am+n = a"a" =o(m)o(n).构造To show that is injective, suppose that m and n are two elements in Z证明where m + n.We can assume that m > n. We must show that am + anLet us suppose the contrary; that is, am = an.In this case am-n =e, wherem-n > O, which contradicts the fact that a has infinite order.Our mapis onto since any element in G can be written as an for some integer n and口d(n)=an
观察 构造 证明 如何判断两个系统的同构？
问题3.1：这个定理给我们什么感觉？Theorem 9.5 The isomorphism of groups determines an equivalence relationontheclass of allgroups.如何去证明这个定理？
问题3.1：这个定理给我们什么感觉？ 如何去证明这个定理？