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 ?

1 : t.dk:1#-Yd)Geometric meaning

¥µ¥ g n:

The collection ofall matrices that

present the like

{y=✗ } c /R2