计算机问题求解一论题4-3一群同态基本定理2019年3月20日
计算机问题求解 – 论题4-3 - 群同态基本定理 2019年3月20日
问题1：我们为什么定义“同构”函数？iso-morphologyTwo groups (G,)and (H,o)are isomorphic if there exists a one-to-oneand ontomap :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“同构”函数 ？ 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？Let o:GH be an isomorphism of two groups. Then thefollowing 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 G has a subgroup of order n, then H has a subgroup of 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？
如何判断两个系统不同构？Example9.5.Even though S3 and Ze possess the same number of elements,we wouldsuspect that they are not isomorphic,because Z is abelian and S3 is nonabelian.Todemonstratethat this is indeed the case, suppose that @:ZgS3 is an isomorphism.Leta,bESsbetwo elements suchthatabba.Since@isanisomorphism,thereexistelementsmandnin Zesuchthatd(m)=aandΦ(n)=bHowever,ab= o(m)o(n)= o(m +n)=Φ(n +m)=Φ(n)(m) = bawhich contradicts the fact that a and b do not commute
如何判断两个系统不同构？