MATH 3231 HWI Youwenzhanglwynn) 31 1 2 1 2 02 1 ⻔ Notethat [ 697 EH Now let [ 台 包 , 7 . [台 名 ] EH Then 珆"⻔ [ 台 贤 ] = [ 㘜 夞⻔ hwwsincea.ci#0andazCzilo=)aiazCiCztoThus绐 '⻔ [ 台 名 ] = [悠悠热2 ] EHAkonotethatforanyacEIRifactothenyactoi.io?II%c1汇出 了 话 !您] = i '。 9 7 ThusHissubgroupGb.liR ) NownotthatforanyzxzmatrixinGlzllRIUi.I.wehaeii.FI= [ 笑T.I-ThusforEEJEHandiii.IE GldlRJIiii.IT?I 点 了 -1 cxwi-t.y.li/XiW1-bxm-y.cz.-axiy.+bx,2+y.x,c , - 2 1 2 5 - 21 WC z.xil.tw. x . C ] lǖtwhich belonqstoHifdeteminantofthismatrixisnon-zeroandz.me _ Z.me#Zib=-WiCWhichisnotalwgstrueThusHisnotanormal subgroup。 1 b) letlfiH-IRXIRAbedefineo.se 治 是 7-la.CI Then det [ 台 只 了 , 沿 是 JEH.thena.ci#0azczt0I!䟓 话 䟭 僽 悠热了 50 4 1 [ 。憇 ] [ 台 趴 = 4 1 㵞 兜! " 了 ) = ( aaz.CI Cz ) = l a i , C ) ˋ ( az.cz ) = 4 1 绐 息 了 ) ˋ 4 4台 怨 了 ) souisqrouphomomorphism.toprovethateisonto.letla.cl EIRAXIRXTheneliqiD-ac.whereoiIEH.sodisont.no c) kerly ) = { [%: 7 E H : 0 1 [% ] ) = 灬 } = { [ 8 EJEHila.cl ⼆ ( 1 1 1 ) } = { [ 吢? ] E H i a = 1 . C = 1 1 = { [IEHibEIRYdet4-kery-clk.tlbedefine.cl by u l [ 公 了 ) = b , b E 1 R - 4 1 [ 。 ' 帄 [ ⻔ ) = 4 1 [ '。 叶 加 J ) = b 、 ⼗ ⼝ 2 = 4 1 [ ? ' ] ) + 4 1[ '。 少 ] ) .la soyisagwuphomomorphism.gtiselearyonto.since.be収 , has foremage [0'7] Eker 141 u u l [ i '⻔ ) = 4 1 ⽐ 个 1 1 Then bi = b 2 =) [ i ' ? ' ] = [ i 个 7 soiyisone-onettenaeisbijective.soyisisomorphi.ms 0 ker 14 ) = 1 1 R , + ) d) nlotethatwepwvedinpartlbl-hatyiH-lixlRtisasuriect.ve qrouphomorphismihws.by firstqroupISOmorphismTheoremHlkereEIRAxlk.lt 2 . letzlajbethecenterofqroupsthenzcajifXEG-xg.iqxforallgEGGIGJProvetr.at ZCGJisanormalsubqroupofGP-iweknowifGic.cn groupandNisanysubqroupofathennlisnormalsubq.roup of G it Nx = x N V X EGMowsinazcai-SXEGixgiqxforaigeayteletziEZIG.Zz.CZ (G ) thenz.g-qzzandzzg-gzzvgc-GNowzz.gg ⇒ Zz " ( Z 2 g ) 2 2 - 1 = Z 2 -1 1 9 2- 2 ) Zi ⇒ lzizz ) 9 2 2 -1 = Eig ) ( 2 2 2- 2- 1 ) ⼆) 9 z i ' = 2- 2 -1 9 [ Zizz = e = Zzzi 7 i. Zz - 1 E Z ( G ) ZzziEZIGJThereforetizzTg-z.czz - 1 9 ) = Z , 1 9 2- z - 1 ) = 1 2- 1 9 ) Z 2 -1 = (9 2 1 ) Z 2 - 1 = glz.ziji.ZZz-IEZIGIHereZIEZIGJ.Zz.EE l a) ThereforezisasubgroupofGMoreaerifq.GG andzEZCGJThengzg-ilgziqt-lzgigtlzg-qzji.tl gg - 1 ) 1 9 9 - 1 = e ) = Z E Z ( G) Thus g = 9 -1 EZIGEgEGandZEZlajtszcGJisanormalsubg.roupot G 2 b) Prouethatif-aisagclicgroup.tnenaisanablean.l-iGiven-isagcucqroup.thenwehave.to proveaisanabeliamqroupif-siscyclicgroupuithq.cnerator xzcGIEeryelementin-canbewr.tnas Xkz for some K E 区 and Z E ZCGI.Nowletg.hEG.tn g = xazandh-xbwforz.WEZIGJ.wehave.gs/=xazxbw=xaxbzw ⼆ xatbzw-xbxazw-xbxawzt.mxbwxaz = xbwxazgh hg axautwxa ⇒ Gisanabltiang.noup _ 3)
欢迎咨询51作业君