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Grumpy • Order of a group 14 = site of a
• order of an element gea
ord (g) =min{ nzl : g"=1 } .
Def Let G be a group ;
a subgroup of G is a subset
HE G which is by itself a group
with the same operation tf C- G
Examples-
① GEG ② { idea③ V4 - Klein 4-group di , a , b. c }
{ 1. a}<V4④ 1R×<¢✗
Rink HEG is a subgroup⇐ IGEH ;
for any gygrc-ttgiogr.tt/,gTc-H
Subgrouplest If HEG ,G-group
H is a subgroup if and only if
G) 11=101(e) for any gihftl goblet
⇐Cs ) It - NOT a subgroup of 2 ?
={ non -nesahk ( g- c- Itintegers }
but -54¥ .
(6) which subgroups of ,# )
do youknow ?
Even numbers II < I
✓ [• non - empty• if g , h are even numbers
"goh" "
g- heels.am even number
Bonus could you hsd all the9-2
subgroups of 2\
teneralDef let G be a group , getthe subgroup generated by g
is the set of all powers of g , ie
(g) = { g"
: n←K }
={ 1. g. g-'
, gigi, - . - . }
9¥ • G-- V4 (a> ={ hate• f- Ci Ci > = { 1 ,i , -1 , -if• G=|R× 627=111214,8114 . .
.
£14181 . - . }
Lenya (g) £ G
PI Use the subgroup test :
• < g) to since it contains g
• if gh , g"
c- (g)(min c- 2)
gmcgy-t-gm.g-w-a.fm -1cg>☒
Lemmata G- group , get
1cg> I = ord (g)
1¥ . Case 1 : ord (g)→
there is no UZI : gh =L
(in this case {gn : not } are
all distinct , since
gn = gm a > m
then J"
= gn.CN)-1=1contradiction .
1cg> I = • .
• Case 2 : ordcg) = n < •
In this use (g) ={ 1 , gig } . . . ,g" }
and these are all distinct
I¥ Any clearest of (g) an be
written as gmm= n - ktr
, Osram
gm= ghktr-cgnkgrn.gr+
Distinct : tf
gm = grin , oerfra.ch
then grt" =Lcontradiction ahu rz-ric.hr
.
☒Det heth be a group
let Gii . . , Grecthe subgroup of a generated by
Gu . . - igr ( notation : Cgi , . . . , gr))
is the set of all elements
which can be written as productsof elements of fogy . . ,gr , gil . . . .gr
")
• If Cgy . . -187=9we say that gin . . . .gr generate f.
( products : gig . gigs-9,95' - " ga )E± . G- V4 (lab> = ?
{ 1 , crib , a.b=c}=Vy
aib generate Vy
.IRX , 111213,4 .
- . . .7--1%0
i¥•←¥i÷ñ=÷÷:*.elements gihi
hah , g-i.si;g-
' high;gh , gli'
, hg , big , big-1 , hg-1
,
ghh , ghgt , . . . .- tight , - . .
.
ghgh" . . - ]
÷:÷÷⇒*order 3
#:#¥aa={;
can a. b= b ?
a. b. b-'= b. b-1 ⇒ a- I
77 *
a. b= a ? ⇒ b=1☒
if a. a-- I = a. ba-
'a.a=a÷ab ⇒ a=b ☒
%
a. a# a since otherwisea-=\
b.to#bCb--Db.b--lCsnhaba--Da-tbsud!:fIee::-.otaappears exactly one in
each row and column ofthe Cayley table .
* Why does g have to appear in
the f-th row of the table ?
g is in the fth row
h - th cot
⇒ g = fh ⇐
f-tg-f-lfh-ha-f.ee¥a'=b a' = 1--09
ik&×2 matrices
G- { 19 ! ) : ad - be _t0i**be }
H= { 111 ) : a+b= ad }ad - bcto
k={ Cc ! ) : ad - be -40, } .ab=cd
which ones are groups ,and which ones are not ?
E (1711331--6%4)
I I 7-not closed under product .
Hw detcghfdet detail
so of detcgidetlhko Atthen also detlghtto
(a b
c d) i (& F) c- H
? ( I 11"
c- H
? KIKI ! )eH1%1-1=1*1!if a+b=ctd
thend-b = - cta
⇒ ÷. - ¥n=-÷t÷⇒ the inverse is in H .
(1111%1)=4"" armed
coitdc died'd)? aoitbdtabtbd
1- can -idctcbtddrearrange
: Lns=a(ñfb)+ bad)
⇒*. !÷÷÷÷÷a-ib-ctd-8%bf.ZS-ds-cc-idif.SI.
The expressions are
equal !
What does it mean
that a+b=c+d ?