; TeX output 2003.09.10:1631ufv6fv홊U9nK`y 3 cmr10AfCLASSIFICAeTIONOFFINITE2-GR!OUPSWITHSOME GNON-METeA!CYCLICfMAXIMALSUBGROUPSANDWITHALLUmSECONDfMAXIMALSUBGR!OUPSBEINGMETeACYCLIC:V.ޟ;jfCepuli"Dc,M.IvdDank!oviDc,fE.Ko!vdDaDc-Striko'UNIVERSITYfOFZA!GREB_CR!OAeTIA;In!troMductionGThe M': 3 cmti10(sepelJanko1,dTh.7.1)Let b> 3 cmmi10Gbeaminimalnon-metacyclic2-6grpoup. vThenGisoneofthefolFlowinggroups:6(a)TheelementaryabpeliangroupEz|{Ycmr88oforder86(b)ThedirpectproductQz8.!", 3 cmsy10nZz26(c)ThecpentralproductQz8.nZz4oforder246(d)G =ha;1b;cja4ʫ= b4=[a;1b]=1;c2=a2b2;a2cmmi8c9 =aK cmsy81 \|;bc9 =a2b3i,6wherpeGisspecialoforder25withexp(G) =4,6 z1(G) =G0=Zȁ(G)=(G)=ha2;1b2il=Ez4;6MMLpet2Gbeasecond-metacyclicgroup.ThenGisoneofthefol- 6lowing17grpoups:G(a)fourgrpoupsoforder16:6Ez16 ;1Zz4.nEz4;Dz8nZz2,orG =ha;1b;cjc4ʫ= 1;bc9 =abil=Ez4nZz4G(b)tengrpoupsoforder32:61)G >Hhlh=}UQz8.nZz2;Hh=ha;1b;cja4ʫ= 1;b2=a2;ab-=a1 \|i:6Gz1ʫ= hHH;1djcdO=a2ci=ha;1binhc;di l =Qz8.Dz86Gz2ʫ= hHH;1djbdO=abc;cd=a2ci6Gz3ʫ= hHH;1djd2= a2;adO=a1 \|;bdO=abi6Gz4ʫ= hHH;1djd2= a2;adO=a1 \|;bdO=bci6Gz5ʫ= hHH;1djd2= c;adO=a1 \|;bdO=abi6Gz6ʫ= hHH;1djd2= ci=ha;1binhdi l =Qz8.Zz46Gz7ʫ= hHH;1djd2= a;bdO=bc;cdO=a2ci62)G>Hl߹=vCQz8S:OZz4;H߹=ha;1b;cja4 "=1;b2=c2=a2;ab )=a1 \|i6andifG >L,thenL  3 msbm10Qz8.nZz2ʫ:6Gz8ʫ= hHH;1djd2= ci6Gz9ʫ= hHH;1djadO=a1 \|;bd=abi6Gz10 = hHH;1djd2ʫ=ac;bdO=abiG(c)thrpeegroupsoforder64:6G >Hh=ha;1b;cja4ʫ= b4=1;c2=a2b2;ac9 =a1 \|;bc9 =a2b3i6Gz1ʫ= hHH;1djd2= a2;adO=a3b2;bdO=b1 \|i6Gz2ʫ= hHH;1djd2= b2;adO=a1 \|;bdO=a2b3;cdO= aci6Gz3ʫ= hHH;1djd2= b;adO=ab2;cdO=aciC32 ufv6fv홊GWeefbMeginb!yrecallingsomebasicde nitionsandfacts.q6De nition21|AgroupľGis> metacyclic܂ifthereexistsacyclicnormalsub- 6groupfNofGwithcyclicfactorgroupG=dDN1.6Theorem23w>MLpet :Gbeametacyclicgroup,THasubgroupofG,TandKԻa6normalsubpgroupofG. vThenHΫandG=Kkarealsometacyclic.6Prpoof: ByDef.1thereexistsN;NG;suc!hthatNGDandG=dDNarebMoth6cyclic.{ByW@akno!wntheoremN1HH=dDNMLpetWWGbeagroup.NIfG=dDZȁ(G)iscyclic,rrthenG=Z(G),rrthat6is,Gisabpelian.6Theorem25w>MLpetGbeanabelianp-group,I jGj=p ,forsomeprimep.d6Then*G l =ZzpV;cmmi6 Aacmr61޾ZzpV ͽ21`Zտp k2; k g ɖu 3 cmex10P 퍑̿i=1S zio= `.\The*k-tuple( z1; z2;1:::l; Ȯk#)鍑6is0uniquelydeterminepduptoorderandsowecanassume z1H1? zi6 zi+1K 1۱ Ȯk#.6De nition22|AnfabMelianp-groupG l =Zzp6In1;nZzp|P`+Z{zP`+Z}.ntimesQ Ezpn iscalledS*6elementaryabpelianp-groupfoforderpnP.6De nition23|LetfGbMeap-group.TheF)rpattini-subgroupofG,6(G) = ɖT |M"MLpetGbeap-groupand(G)itsF)rattini-subgroup. vThen:61)(G)nEGandG=(G)iselementaryabpelian62)Ifj G=(G)j=pnP,thenGisgenerpatedby(atlepast)nelements63)F)ora2-grpoup,(G) =fz1(G):=hx2jx2Gi6De nition24|Let gK2tG;;SJG.ThesetSgH:=gd1 SgK=fgd1sgJj s2Sg6iscalledtheg-cponjugateofS,andthemapping'zgߥ: G!G;Y&'zg:x7!xg=6gd1 xg isfcalledthecponjugationbyg.C33ufv6fv홊6Theorem27w>MThekcponjugation'zg:x7!xg @iskanautomorphismofG,'that 6isG'zg c=G;@x'zg=yd'zg,xg=ydg ))x=yd;@x'zg'gI{q% cmsy60 ]=x'gI{g0 û,thatis:6(xg)gI{-:0 l= xgI{g-:0 û.6De nition25|LetVS$5G;[HfG.TheƻnormalizerofS\XinHistheset6NHD(S)@h:=fh2Hݜj۾Sh G=Sg,Nandthe#6cpentralizerofSȻinHistheset6CHD(S) :=fh2H'jf8s2S;yshp=sg.Ob!viouslye,shp=s,sh=hs.6Theorem28w>MLpetS9 G;HhG. vThen:61)NHD(S) G;CH(S)GandCH(S)nENH(S):62)dF)orK7G,DthefactorgrpoupNHD(Kȁ)=CH(K)d̻isisomorphictosome6subpgroupofAutKk-thegrpoupofalFlautomorphismsofKȁ.6Remarks :61)fK7mEnNG(Kȁ).62)ECG(G)p=fz2Gj8gd;vgz=pz{IggZȁ(G)E-thecpenter(ofGisthegroup6consistingfofthoseelemen!tsinGwhichcommutewithallelementsofG.6De nition26|Thej6c!hainG =Gz0BGz1B1{BGzi17BGzi[fB1BGȮk.9= 1j6isa6cpompositionseries-ofͶG,׊ifeac!hGzi2isamaximalnormalsubgroupofGzi1AV.6If,moreo!ver,eachZGziU4shouldbMeamaximalsubgroupofGzi11normalinG,6suc!hfaseriesiscalledachiefseriesoffG.6Theorem29w>MLpetGbeap-group. vThen61)Zȁ(G) >162)ItisNG(PV) >Pp@foranyP< G. vEsppecialFly,ifjG:PVj=p,thenPBCnG.63)=A\lFlfactorgrpoupsGzi1AV=Gziofacompositionseriesorachiefseriesareof6orpderp,jGzi1AV=Gzidj =p.64)sEachsubpgroupsPɻofGbpelongstosomecompositionseries,andeach6normalsubpgroupNofGtosomechiefseries.6Theorem210}!Lpetl!E beanelementaryabeliansubgroupofa2-groupG,}and6go:2 G;gd2/>2E. vThenjCE-(gd)j2ʫjEj.6Prpoof: ԹBecause]ofgd2 _2;EandEabMelian,xgI{-:2 :Ĺ=;xforan!yx2G.Th!us6(xxg)g ù=žxgxgI{-:2 j=xgx=xxg,>forallx2G,>andsoxxg 2CE-(gd).No!w6xxg G=Iydyg !,xyLܹ=xgyg !ڹ=(xy)g G,xyL2CE-(g),CE-(g)x=CE-(g)y.6Therefore.xxgH6=-Jydygf,CE-(g)x6=CE-(g)y,`and.sojCE(gd)j-JjEܹ:CE(gd)j)6jCE-(gd)j2ʫ jEj.C34%ufv6fv홊a2-GR!OUPSfOFORDER 24#6Theorem211u LpetGbea2-groupoforder 24. vThenoneofthefolFlowingholds: 6(a)F)orjGj =2:G=haja2ʫ= 1il=Zz26(b)F)orjGj =4:6Gz1ʫ= haja4= 1il=Zz4orGz2=ha;1bja2= b2=1;ab-=ail=Ez46(c)F)orjGj =8:6Gz1 ]=ha)ja8=1il=(Zz8;ZGz2=ha;1b)ja4=1il=(Zz4 Zz2;ZGz3=6ha;1b;ci|l|=>Ez8;+qGz4 `=|ha;bX@ja4=|1;+qab X=a1 \|il=>Dz8 D-X@dihepdralgroup,6Gz5ʫ= ha;1bja4= 1;b2=a2;ab-=a1 \|il=Qz8-quaterniongrpoup6(d)F)orjGj =16:G( `)abpeliangroups:6Gz1ʫ= haja16 =1il=Zz16 ,Gz2ʫ=ha;1bja8ʫ=1il=Zz8.nZz2,6Gz3ʫ= ha;1bja4= b4=1il=Zz4.nZz4,Gz4=ha;1b;cja4= 1il=Zz4.nEz4,6Gz5ʫ= ha;1b;c;dil=Ez16G( )1nonabpeliangroups,exp=4GYd=81-containingacyclicmaximalsub-6grpoup: vGz6ʫ= ha;1bja8= 1;ab-=a1 \|il=Dz16 -dihepdralgroup,6Gz7ʫ= ha;1bja8= 1;b2=a4;ab-=a1 \|il=Qz16 -quaterniongrpoup,6Gz8ʫ= ha;1bja8= 1;ab-=a3il=SDz16 -semidihepdralgroup,6Gz9ʫ= ha;1bja8= 1;ab-=a5il=Mz16 -M-grpoup.G( )nonabpeliangroups,expG =4:R1)G >Ll=Ez8:6Gz10 = ha;1b;cja4ʫ=1;ab-=a1 \|il=Dz8.nZz2,6Gz11 = ha;1b;cX{jc4ʫ=1;+bc9 =abi=ha;1bi0hcil=Ez4Zz4,utheX{semidirpectproduct6ofEz4byZz4.Ra2)˾G>L)LEz8.Then,byTh.8andTh.9,therpeissomeK:=CqG,6jKȁjW̹=4.If0 K Ml M=_Zz4,mtheorpderjAutKjW̹=20 andifK Ml M=_Ez4,mtheorpder6jAutKȁj =6.AThusLjNG(K)=CG(K)j 2,andLtherpeexistssomeabeliansub-6grpoupCHh< G,Hhlh=}UZz4.nZz2and9x2GnHH;x2ʫ=1:6Gz12 =ha;1b;cgja4ԝ=1;Øb2=c2=a2;Øab7u=a1 \|i=ha;1birhcil=xQz82Zz4,the6cpentralproductofQz8andZz4.V2b)G >Hhlh=}UZz4.nZz2andx2GnHh)x2ʫ6=1,thatisjxj=4:6Gz13 = ha;1b;cja4ʫ=1;b2=a2;ab-=a1 \|i=ha;1binhcil=Qz8.Zz2,6Gz14 =ha;1bja4=b4=1;Sab<=a1 \|i=haikhbil=Zz4oZz4,thesemidirpect6prpoductofZz4byZz4.C354ufv6fv홊>PR!OOFfOFTHETHEOREM2ݍGA.fGroupsoforder16. 6AsOgroupEz8PSistheonly(minimal)nonmetacyclicgroupoforder8,thesec-6ondv.metacyclicgroupsoforder16arethoseamongthemwhic!hcontainEz8.6According?toTh.11(d),5w!ehavefoursuchgroups:Zz4i}Ez4;pEz16 ;Dz8}Zz26andfEz4.nZz4,thesemidirectproMductofEz4fjb!yZz4.GB.fGroupsoforder32.6AccordingCtoTh.1,suc!hagroupcontainsasubgroupisomorphictoQz8ԬZz2,6orftoQz8.nZz4,thecen!tralproMductofQz8fjbyZz4.J,B1.tG >Hh=ha;1b;c,ja4ʫ= 1;]b2=a2;]ab-=ai=ha;1biFxhcil=Qz8|Zz2.6No!w,zRjG߹:Hj=2,G=hHH;1di,d2 2H. JSince(H)=fz1(H)=6hx2 *jX&xݏ2Hi=ha2iand z1(H)ݏ=hx2H;jx2 =1i=ha2;1ci,Ėth!us6ha2i; |ha2;1cic!harMKH1 andsoha2i;ha2;1ci/CG.ҍTheMKmaximalsubgroupsof6Harethefollo!wingones:,ha;1bil=rpha;bcil=hac;bil=hac;bcil=Qz8,and6ha;1ci l =hb;ci l =hab;ci l =Zz4.nZz2.6Wee4usethebarcon!vention4forsubgroupsandelemen!tsoffactorgroups.For郍64p 䍾GA= G=ha2i,w!eJhave4p 䍾H۹= HH=ha2i=hp ɾNaɾ;1p   b;;1p NcKil=Ez8,andJ4p 䍾G=h4p 䍾H ;1p  d8i;{p  d r_a2=-24p 䍾H .6ByfTh.10,itisjClj\)DHD(p  d)j2ʫ jHj=8,fandsojClj\)DH(p  d)j 4.6Ontheotherhand,Pha2;1ci55CGimplieshp Nci55C4p 䍾GZ¹andp Nc QK2 Clj\)DHD(p  d).>AsjClj\)-E-(p  d)55\6hp ɾNaɾ;1p   b;ij2,[some^oftheelemen!tsp ɾNa ;Տp   b ,orp ɾNa p   biscon!tainedinClj\)-E-(p  d),andw!e6canfassumewithoutlossthatp ɾNa z2 Clj\)-E-(p  d).z^6No!w,fp   bYp򼭉\)ߨdC2 4p 䍾H nhp ɾNaɾ;1p NcKi =hp ɾNa;1p NcKinp   b!,fandso:6G =hHH;1dfjd2ʫ2 H;yadO=azz1;bdO=a"lоc hbzz2;cdO=czz3i;6";1o:2 f0;1g;yzz1;zz2;zz3ʫ2 f1;a2g.6Therefare3caseswithrespMecttotheelemen!td.G1)f9d 2GnH,s.th.ݾd2ʫ=1.6Since;Eha2;1c;diEz8Iitm!ustbMecdl6=c,`}andsocdl=a2c.zIfadl=a3,`}then6(ac)dé=adߨcd=a3a2c=ac,Ijand(replacingawithac,w!ehavewithoutloss6adO= a;ycd=a2c.6No!w zbd-:2fj=\b1 =b=(bdߨ)d ;ù=(a"lоc hbzz2)d=a"lоa2 Mlc ha"cbzz2zz2 =\a2("+I{)s,b.6Thereforef" =o:=0or"=o:=1.6If"l==0,;thenbd n=bzz2.Feorzz2 Np=a2 Oitis(bc)d n=ba2a2cl=bc,;and6replacingfbcwithb,w!ehavewithoutlossbdO= b.ThusݍYݾGz1ʫ= hHH;1dfjczdO=az2ci:C36Iufv6fv홊6If#$"ڍ=? =1,BSthenbd5=acbzz2=azz2bc.TNo!w,BSreplacingawithazz2,BSw!eget 6withoutflossbdO= abc,andthegroup:SdGz2ʫ= hHH;1dfjbzdO=abc;yczd=az2ci:G2)fx 2GnHh)x2ʫ6=1;y9d2GnH's.th.ݾd4=1:6No!wwd2 2H[bandd2 7isaninvolution,d2 =fa2;1c;a2cg. QAswa2candc6are.`in!terchangeable,^wemayassumethatd2 W2fa2;1cg. uIfd2=c,^then6cd=N(d2)d=d2R=c.Ifd2=a2,$then(cd)2=cd2cd=ca2czz2R=a2zz3R6=1,$b!y6ourfassumption.Th!uszz3ʫ= 1,andsocdO=cinbMothcases.Jf2a)fCased2ʫ= a2:6No!w,Z(ad)27=3ad2ad۹=a3azz17=zz16=1,Zb!y6ourassumption.Thuszz17=3a26andfsoadO= a3;ycd=c;yd2ʫ=a2.6FeorB"s=1,jbd=ac hbzz2.NReplacingawithazz2,jw!emayassumebd=sac hb.6IfȾܹ=I1;bd{=acb,andreplacingawithac,w!egetbd{=Iab,asinthecase6o:= 0,fandsoGz3ʫ= hHH;1dfjdz2= az2;ybzdO=abi:(6Feor"w=0,.itisbd=c hbzz2.#TIfzz2{=a2,.thenbad "=(a2b)d=a2c hba2{=cb.6Replacingdwithad,#Nw!egetbdO= c hb.>Feoro:=0;չbdO=band(bd)2ʫ=bd2bdO=6ba2b =1,facon!tradiction.Thus,fo:= 1,bdO=bc,andw!ehave:5Gz4ʫ= hHH;1dfjdz2= az2;yazdO=az1 \|;bzdO=bci:Jf2b)fCased2ʫ= c:6Feorf" =1;9bdO=ac hbzz2,sIandreplacingawithazz2,sIbdO= ac hb.ȑAgainifo:=1,6replacingwawithac,w!egetbd=oab.Now,bd-:2 %=b=(bdߨ)d=(ab)d=adߨbd=6azz1ab )zz1ʫ=a2,fandso:&7Gz5ʫ= hHH;1dfjdz2= c;yazdO=az1 \|;bzdO=abi:6Feor"=0;,bd =c hbzz2.If=1,3_thenbd =cbzz2,3_and(bd)2 =bd2bd =6bccbzz2й=Q̾b2zz2=a2zz26=1,b!yourassumption.]Thuszz2й=Q1and(bd)2=Q̾a2,6whic!hAleadstothecase2a). 6Thuswemayassumethat=d0,andso6bd=`bzz2; ad=azz1; cd=c:8Th!us(adߨ;1bd)2f(a;1b);(a;b3);(a3;b);(a3;b3)g.6Aspa;1bandabarein!terchangeablephere,{andforadO= a3;Dbd=b3ʫ)(ab)d=6a3b3ɹ=žab,7=thereFremain,withoutloss,onlyt!wocases:Ŝadm=ža;wbd=bFand6adO= a;ybd=b3.6Inthelattercase(ad)2.=nad2adM=aca=a2c;Aaad ю=adM=a;bad ю=(b3)dM=6b9= b;yٹ(a2c)ad n=(a2c)d꽹=a2c,andreplacingcwitha2c,anddwithad,w!e6getfwithoutloss,thatadO= a;ybd=b;ycd=c,fandth!us3Gz6ʫ= hHH;1dfjdz2= ci:C37^ufv6fv홊G3)fd 2GnHh)jdj=8 6No!w,d2ݣ2Handjd2j=4.Asallelemen!tsoforder4inHareinterchange-6able,_w!eMmayassumethatd2ʫ= a,_andsoadO=a.UNo!wd2ʫ=a; adO=a;bdO=6a"lоc hbzz2;cd O=%czz3.Ifξ"=1,:then(bd)2 嫹=bd2bd O=baac hbzz2=c hzz2,:an6in!volution,against*ourassumption.(Therefore"B=0,and*bdf=c hbzz2.(No!w6bd-:2 8=ba =b3=(bdߨ)dݑ=(c hbzz2)d=c hz<{Iv3cbzz2zz2=z<{Iv3b)z<{Iv3 =b2=a2)6zz3|)=%a2; =1.[Th!usbd͹=bczz2;cd͹=a2c;ad͹=a:Ifzz2|)=%a2,+replacingc6withfa2c,w!egetbdO= bcand nally:8Gz7ʫ= hHH;1dfjdz2= a;ybzdO=bc;czdO=az2ci:JfB2.ݾG >Hh=ha;1b;cfja4ʫ= 1;yb2=c2=a2;yab-=a1 \|il=Qz8.nZz4; 6G =hHH;1di;yd2ʫ2H.No!wffz1(H) =(H)=ha2i;yZȁ(H)=hci:6There0are8elemen!tsoforder4:a;a3;b;a2b;ab;a3b;c;a2c,Vand076in!volutions:ha;1b;di\?=ha;1bihdi\?l\?=<ľQz8OZz2,against[theassumption.pG6isfisomorphictoGz1.61b)fx 2GnHh)x2ʫ6=1;y9d2GnHH;jdj=4:6No!w,۾d2ais]aninvolutiononH.Weemayassume,withoutloss,thatd2=Pa26orfd2ʫ= ac.6IfvȾd2 %=ea2,then(ad)2=ead2ad E=aa2azz1 %=zz16=1;I(bd)2=zz26=1,and6(abd)2ʫ= zz1zz26=1,facon!tradiction.6Ifgbd2ʫ= ac,sthenbd-:2 =bac =b3ʫ=(bdߨ)dO=(bzz2)d=bzz2zz2ʫ=b,sagbcon!tradiction6again.61c)fx 2GnHh)jxj=8;yd2GnHH;d2ʫ2H:6Weefma!yassume,withoutloss,thatd2ʫ= a,ord2= c.C38 rufv6fv홊6Iffd2ʫ= a;ybd-:2 =(bzz2)dO=b=bav=b3,facon!tradiction. 6Th!usnd2ʫ= c;B adO=azz1;bdO=bzz2;cdO=c.v No!wn(adߨ;1bd) 2f(a;b);(a;b3);(a3;b);(a3;b3)g6Inthelattercase(ab)d7=X=a3b3A=ab,andsincea;1bandabma!ybMereplaced6with eac!hother,4swemayassumethat:&ad= a;;bd=b orad=a;;bd=b3.6Inthelattercase(ad)2ʫ= a2c;sbad n=b,andth!usreplacingcwitha2candd6withfad,thesecondcaseisreducedtothe rst,andw!egetfGz8ʫ= hHH;1dfjdz2= ci=ha;1binhdi l =Qz8.Zz8:GCasef2): 6Replacingfawithazz2,w!emayassumethatbdO= ab.62a)f9d 2GnHH;yjdj=2:6ha2;1ac;di Ez8ʫ)(ac)dO=aczz1zz3ʫ6=ac)zz36=zz1.JIfeadO=a,rthenbd-:2 =b=6(bdߨ)dO= (ab)d=aab=b3,facon!tradiction.ThereforeadO=a3;ycd=c,fandM4Gz9ʫ= hHH;1dfjazdO=az1 \|;ybzd=abi:62b)fx 2GnHh)x2ʫ6=1;y9d2GnHH;jdj=4: 6No!w,fwithoutlossd2ʫ= a2fjord2= acord2= bc.6Ifad2]L=Ha2,_then(ad)2=Hzz16=1aand(cd)2=Hzz36=1,_th!usazz1=zz3=a2:aSo6G =hHH;1dfjd2ʫ= a2;yadO=a3;ybdO=ab;cdO=c3i:6HereW(ac)dO= a3c3ʫ=ac,$andG>ha;1d;aci =ha;1di*haci l =Qz8Zz2,$against6thefassumption.Actuallye,G l =Gz2.6If'd2 =ac,Wthenbd-:2 6=(ab)d ˏ=azz1ab=bac /=b3,Wand'sozz1 =1.@ No!w,6(ac)dO= ac=adߨcd=aczz3,fth!usalsozz3ʫ= 1.Therefore:Gz10 = hHH;1dfjdz2ʫ=ac;ybzdO=abi:6Ifd2)'=i#bc,Cthen(bc)dH˹=bc=abczz3)'=azz3bc,Cimplyingazz3)'=1,Cacon!tradic- 6tion.62c)fx 2GnHh)jxj=8;yd2GnHH;d2ʫ2Hh:6Weefma!yassume,withoutloss,thatd2ʫ= aord2= bord2= c:6Feorld2ʫ= a,xbw!eget(bd)2= bd2bdO=baab=1,xbalcon!tradiction.ʱAsbd= ab,xbit6cannot؞bMed2]=^Yb.tIfd2=^Yc,+then(bd)2=^Ybcab=b2cab5=a2ca1չ=ac,+and6sofjbdj =4,acon!tradictionagain.C39 ufv6fv홊GC.fGroupsoforder64. 6AccordingotoTh.1(d),suc!hagroupGcontainsasubgroupisomorphicto6thegroupH2=qha;1b;cja4 u=b4=1;c2=a2b2;ac =a1 \|;bc =a2b3i,6where z1(G)=hx2Gjx2 W=1i=Zȁ(G)=(G)=ha2;1b2i4l= Ez4.6One~~caneasilyc!heck~~thatthereareonly4squareroMotsfora2>(thatis,ysuc!h6xE2G,Rthat"x2 =a2),and"12squareroMotsforb2 &anda2b2eac!h.Thus6A =ha2ic!harG.«TheTsquareroMotsofa2չgeneratethesubgroupL =ha;1b2il=6Zz451Zz2.ThegroupM=Sha;1biistheuniquesubgroupofH}isomorphic6toqZz4oZz4.Th!us,A;1K,;L;M9areallc!haracteristicinH2andconsequently6normalfinG,asHRCnG.6Weefha!ve64p 䍾GC@ٹ=־G=ha2i>4p HG=hp ɾNaɾ;1p   b;;1p Nc ,jp ɾNa fA2=p   b W_a4=p y^1 p;op Nc2=p   b W_a2;op ɾNa 9r|\).yBc =p ɾNa n;op   b "\).yBc =p   b W_a3i=6hp   b ;1p Nc 6jfp   bYp_a4$= p y^1A;yp Nc2\= p   b_a2 };yp   b ,\).yBce= p   b_a3inhp ɾNaɾi l =Qz8.Zz2.6Eac!hsecondmaximalsubgroupofGismetacyclicandbyTh.3thesame6holdsq0trueforeac!hsecondmaximalsubgroupof4p 䍾G 3.> 3 cmmi10Aacmr6|{Ycmr8': 3 cmti10"V 3 cmbx10K`y 3 cmr10: