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

= 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)

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