27

9 3 21 ColleetHW

Last time showed that if S has an identity element it'sunique

the let S l be a bin structure w identity elf e let s l be

another binarystructure Assume thereis a ftu d s s satisfying

thehomem property and assume d is surjective onto Then Ole is an

identity element for shence the identityelement for s

if ur x e s Wis del x x te del x

Assume x 0 x where x is some il't of S ok since 0 is out

Ole x del out Ole x 01 1 x

The other is thesame11

See examples at end of 53

4 GroupFormalize some propertiesof R or I or Q or I that we

take for granted

28

Ex Solve for x x 3 8

x 3 G 8 1 3 add 3 to bothsides

7 31 63 5 computation

7 3 61 5 assoc3 is theadditiveinverseof 3

y 5 property of 0

thisworksble Rt is agaspmi partie

G'G G

Def Aged G l is a binary structure satisfy

Cal f a b c e G Ca b c a btc assoc of

b J ee G st axe e a a fate identity elf

c foreach a EG 7 a e G st aka a a pinverses

I will bechangingthisnotation shortly both

and C

Note that even commutativityisn't assumed

Def A group is abelian if is commutative

Ey U all IR t and Z t abel

Un abel Rt all

abel Maciel t all

t abd Ghar not all

where Gh R is

29

E frombookI

let G at

Define a b by a tb of

Clan Cat is a group

a check assoc catb c e at cl last 94 a I

a b ax b c

h identify elf is 2

at 2 4 a 2 a

c Given a a Qt check a I theinverseelement

ax I I n tReward Don't gettoo dependent

on whatyouknow for familiar objects e.g

IR Q E We have a set a binaryoperation and a coupleofaxioms and

we want to know what we can say about it

Propertifgroups

the identity elt is unique just saw

the Clift rightcancellation

Gt G l be a group let a b c e G

If a b at c ther b c

n If b a c a then b c

30

pHlet a kegu

inverseof a

a b a c

a atb a ate

a a b Ca al c

ex b e c

b c

Same for ight cancellation X

TI let G l he a group let a be G Then

la the beer egg at x b has a soln namely x a b ohne a

is the inn of ab thesolution is uniquec Simbly y a b has a unique solufor y

pfal x a b is g solution plugin

b Uniqueness Say x Xz are solusThen a x b a z Leftcancellation

o samefor y a b k

thy for each at G its inverse a is unique