凯莱定理证明_皮克定理证明

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Theorem1.Let n be a positive integer, and let G = be a cyclic group of order n.Then G≌(Zn;+).Consequently, any two cyclic groups oforder n are isomorphic to each other.Theorem2.Let G = be an innite cyclic group.Then G≌(Zn;+).Consequently, any two innite cyclic groups are isomorphic to each other.Theorem3(Caycley's Theorem).If G is a group,then G is isomorphic to a subgroup of SX, thesymmetric group on a set X(it is called a transformation group(交换群)of X).In particular,if G is fnite, then G is isomorphic to a subgroup of Sn(it is called a permutation group(置换群)).定理6.2(Cayley定理)任何一个群都与某个变换群同构.证明设G是群.对于每一个aG,定义G的变换σa如下:σa(x)ax,xG.显而易见,σa是G的一一变换.令G'{σa|aG}.下面我们来阐明G'是G上的一个变换群.事实上,显然,我们有IGσeG'.此外,对于任意的σa,σbG',我们有(σaσb)(x)abxσab(x),(σaσa1)(x)aa1xxIG(x),(σa1σa)(x)a1axxIG(x),xG,从而,σaσbσabG',σaσa1σa1σaIG.所以G'是G上的一个变换群.现在考察由下式定义的G到G'的映射f:f(a)σa,aG.显而易见,f是满射.对于任意的a,bG,我们有f(a)f(b)σaσbσa(e)σb(e)ab.因此f是单射,从而,f是双射.此外,我们有f(ab)σabσaσbf(a)f(b),a,bG.所以f是G到G'的同构,从而,GG'.Pf:AumeG is a group.For everyaG,we have

σa(x)ax,xG.It is obvious thatσais a transformation of G.LetG'{σa|aG}.obviously,we have IGσeG'.Otherwise,for everyσa,σbG',we have

(σaσb)(x)abxσab(x),(σaσa1)(x)aa1xxIG(x),(σa1σa)(x)a1axxIG(x),xG,thus,σaσbσabG',σaσa1σa1σaIG.Therefor G'is a transformation group ofG.Now inspect thef by type definition G toG' :f(a)σa,aG.Obviously,f is a surjection.For alla,bG,we have

f(a)f(b)σaσbσa(e)σb(e)ab.Therefore f is injection,Thus,fis a bijection.Moreover,we have

f(ab)σabσaσbf(a)f(b),a,bG.So fis isomorphic fromGtoG',thus,GG'.