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CHAPTER9NormalSubgroupsandFactorGroupsNormalSubgroupsIfHG,wehaveseensituationswhereaH6=Ha8a2G.Definition(NormalSubgroup).AsubgroupHofagroupGisanormalsubgroupofGifaH=Ha8a2G.WedenotethisbyHCG.Note.ThismeansthatifHCG,givena2Gandh2H,9h0,h002H3ah=h0aandah00=ha.andconversely.Itdoesnotmeanah=haforallh2H.Recall(Part8ofLemmaonPropertiesofCosets).aH=Ha()H=aHa1.Theorem(9.1—NormalSubgroupTest).IfHG,HCG()xHx1✓Hforallx2G.Proof.(=))HCG=)8x2Gand8h2H,9h02H3xh=h0x=)xhx1=h02H.ThusxHx1✓H.((=)SupposexHx1✓H8x2G.Letx=a.ThenaHa1✓H=)aH✓Ha.Nowletx=a1.Thena1H(a1)1=a1Ha✓H=)Ha✓aH.Bymutualinclusion,Ha=aH=)HCG.⇤117
1189.NORMALSUBGROUPSANDFACTORGROUPSExample.(1)EverysubgroupofanAbeliangroupisnormalsinceah=haforalla2Gandforallh2H.(2)ThecenterZ(G)ofagroupisalwaysnormalsinceah=haforalla2Gandforallh2Z(G).Theorem(4).IfHGand[G:H]=2,thenHCG.Proof.Ifa2H,thenH=aH=Ha.Ifa62H,aHisaleftcosetdistinctfromHandHaisarightcosetdistinctfromH.Since[G:H]=2,G=H[aH=H[HaandH\aH=;=H\Ha=)aH=Ha.ThusHCG.⇤Example.AnCSnsince[Sn:An]=2.Note,forexample,thatfor(12)2Snand(123)2An,(12)(123)6=(123)(12),but(12)(123)=(132)(12)and(132)2An.Example.hR360/niCDnsince[Dn:R360/n]=2.Example.SL(2,R)CGL(2,R).Proof.Letx2GL(2,R).Recalldet(x1)=1det(x)=(det(x))1.Then,forallh2SL(2,R),det(xhx1)=(det(x))(det(h))(det(x))1=(det(x))(det(x))1(det(h))=1·1=1,soxhx12SL(2,R)=)xSL(2,R)x1✓SL(2,R).ThusSL(2,R)CGL(2,R).⇤
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