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1、I.7GeneratingSets1SectionI.7.GeneratingSetsandCayleyDigraphsNote.Inthissection,wegeneralizetheideaofasinglegeneratorofagrouptoawholesetofgeneratorsofagroup.Remember,acyclicgrouphasasinglegeneratorandisisomorphictoeitherZ(ifitisofinfiniteorder)orZn(ifitisoffinite
2、order),byTheorem6.10.However,therearemoregroupsthanjusttheoneswhicharecyclic.Example7.1.RecalltheKlein4-group,V:∗eabceeabcaaecbbbceaccbaeThentheset{a,b}issaidtogenerateVsinceeveryelementofVcanbewrittenintermsofaandb:e=a2,a=a1,b=b1,andc=ab.WecanalsoshowthatVisg
3、eneratedby{a,c}and{b,c}.Inaddition,{a,b,c}isageneratingset(thoughwecouldviewoneoftheelementsinthisgeneratingsetasunnecessary).Exercise7.2.FindthesubgroupofZ12generatedby{4,6}.Solution.Wegetallmultiplesof4and6,sothesubgroupcontains0,4,8,and6.Wegetsumsof4and6:4+
4、6=10.Also,2≡10+4(mod12)=4+4+6.Sothesubgroupis{0,2,4,6,8,10}.Ofcourse,wecannotgenerateanyoddelementsofZ12.I.7GeneratingSets2Note.Thefollowingresultgoesinalittlebitofadifferentdirectionintermsofsubgroups.Theorem7.4.TheintersectionofsomesubgroupsHiofagroupGfori∈Ii
5、sagainasubgroupofG.(Note.SetIiscalledanindexsetfortheintersection.Ingeneral,theindexsetmaynotbefinite—itmaynotevenbecountable.Nowfortheproof.)Note.Foranyset{ai
6、i∈I}withai∈G,thereisatleastonesubgroupofGcontainingallai(namely,theimpropersubgroupG).Soiftheintersec
7、tionofallsubgroupsofGcontaining{ai
8、i∈I}istaken,asubgroupofGcontaining{ai
9、i∈I}results(calledthe“smallestsubgroupofGcontaining{ai
10、i∈I}”).Thisjustifiesthefollowingdefinition.Definition7.5.LetGbeagroupandletai∈Gfori∈I.ThesmallestsubgroupofGcontaining{ai
11、i∈I}isthesubg
12、roupgeneratedbytheset{ai
13、i∈I}.ThissubgroupisdefinedastheintersectionofallsubgroupsofGcontaining{ai
14、i∈I}:H=∩i∈JHjwherethesetofallsubgroupsofGcontaining{ai
15、i∈I}is{Hj
16、j∈J}.IfthissubgroupisallofG,thentheset{ai∈i∈I}generatesGandtheaiaregeneratorsofG.Ifthereisafinites
17、et{ai
18、i∈I}thatgeneratesG,thenGisfinitelygenerated.I.7GeneratingSets3Note.Thefollowingresultshowshowtheelementsofagrouparerelatedtothegeneratingset.Theorem7.6.IfGisagroupandai∈Gfori∈
