南京大学：《计算机问题求解》课程教学资源（PPT课件讲稿）群同态基本定理与正规子群

计算机问题龙解一论题3-18 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-morphology Two groups (G,)and (H,o)are isomorphic if there exists a one-to-one and onto map GH such that the group operation is preserved;that is, (a·b)=(a)o(b) 同构其实可在 for all a and b in G.If G is isomorphic to H is called an isomorphism. 任何代数结构 (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.Then the following statements are true. 1.-1:HG is an isomorphism. 2.G=H. 3.If G is abelian,then H is abelian. 4.If G is cyclic,then H is cyclic. 5.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 G be a cyclic group with infinite order and suppose that a is a generator of G.Define a map:ZG byo:n a".Then 观察 o(m+n)amtn a"a"o(m)o(n). 构造 To show that o is injective,suppose that m and n are two elements in Z, where m n.We can assume that m >n.We must show that am a". 证明 Let us suppose the contrary;that is,am=a".In this case am-=e,where m-n >0,which contradicts the fact that a has infinite order.Our map is onto since any element in G can be written as an for some integer n and o(n)a". ▣

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 rela- tion on the class of all groups. 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？

几个有趣的同构结论 Theorem 9.3 If G is a cyclic group of order n,then G is isomorphic to Zn. Corollary 9.4 If G is a group of order p,where p is a prime number,then G is isomorphic to Zp. Theorem 9.6(Cayley)Every group is isomorphic to a group of permu- tations

几个有趣的同构结论

Carley定理的证明 Theorem 9.6(Cayley)Every group is isomorphic to a group of permu- tations. ·从任意一个群G出发，构造一个置换群G’: ·由置换函数组成的群 G={g:g∈G} ·由G出发，构造置换函数，置换函数的个数和群G相同 入g(a）=ga. ·构造群G到置换群G’的同构函数 p:g→入g ·证明这个函数的双射 ·证明这个函数是G到G’的同构

Carley定理的证明 • 从任意一个群G出发，构造一个置换群G’： • 由置换函数组成的群 • 由G出发，构造置换函数，置换函数的个数和群G相同 • 构造群G到置换群G’的同构函数 • 证明这个函数的双射 • 证明这个函数是G到G’的同构

问题4：为什么下面的结论不叫“定理”？ Proposition 9.7 Let G and H be groups.The set Gx H is a group under the operation (g1,h1)(g2,h2)=(91g2,hih2)where g1,92 EG and h1,h2EH. Example 9.The group Z2,considered as a set,is just the set of all binary n-tuples.The group operation is the "exclusive or"of two binary n-tuples. For example, (01011101)+(01001011)=(00010110) 问题5.1：这个符号是什么意思？ 问题5.2：这个操作从何而来？

问题4：为什么下面的结论不叫“定理”？ 问题5.1：这个符号是什么意思？ 问题5.2：这个操作从何而来？

不难理解的几个定理： Theorem 9.8 Let (g,h)G x H.If g and h have finite orders r and s respectively,then the order of (g,h)in Gx H is the least common multiple of r and s. Corollary 9.9 Let (91,...,9n)EIIGi.If gi has finite order ri in Gi,then the order of (g,...,gn)in IIGi is the least common multiple of ri,...,rn. 问题6：如果诸ri互素，会有什么结论？

不难理解的几个定理： 问题6：如果诸ri互素，会有什么结论？

以下几个结论，余味袅袅 Theorem 9.10 The group Zm x Zn is isomorphic to Zmn if and only if gcd(m,n）=1. Corollary 9.11 Let n1,...,nk be positive integers.Then k ΠZn,≥Zn1mk i=1 if and only if gcd(ni,nj)=1 for if j. Corollary 9.12 If m=p…pt, where the pis are distinct primes,then Zm￥Zp71X…XZp

以下几个结论，余味袅袅