binary operation

Definition 1 – 20.
Let n be a positive integer , the collection Zn is defined as
Zn = { [0] , [1] , [2] , … , [n-1]}.
Definition 1 – 21 .
A binary operation +n defined on Zn as
[a] +n [b] = [a + b ] for each [a] , [b] ? Zn .
Theorem 1 -11 .
For each positive integer n , ( Zn , +n) forms group .
Proof.
H . W .
Note .
( Zn , +n ) called as the group of integers modulo n .
Definition 1 -22.
A permutation of the set N is meant any one – to – one function of
N onto N . The set of all permutation of the set N is denoted by Sn.
1 2 3 . . . n
f(1) f(2) f(3) . . . f(n)

Example .
If N = { 1 , 2 , 3 } , then there are 6 permutation in S3 .
Theorem 1 -12 .
( Sn , o ) is group .
Proof .
H . W .
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Definition 1 -23 .
In any group ( G , ? ) , the integral power of an element a? G
are defined by
ak = a ? a? …? a (k factors ),
a0 = e ,
a-k = ( a- 1 )k,
where k? Z+
Theorem 1 – 13.
Let ( G , ?) be group , a ?G , and m , n ? Z , then
1)an? am = an+ m = am ? an ,
2) (an ) m= anm = ( am)n ,
3)a-n = ( a n)-1 ,
4)en = e
Proof.
It is easy .

Definition 1 -24 .
Let ( G ,? ) be an arbitrary group . For a fixed element a ?G , define
the LEFT – MULTIPLICATION FUNCTION fa : G ? G by
fa ( x ) = a ? x for each x ? G .
Exercise .
Show that the left - multiplication function is one–to–one and onto.
Proof.

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Let x ? G , then
x = (a ? a-1 ) ? x
= a ? ( a -1 ? x )
= fa (a-1 ? x )
So fa is onto .
If x , y ?G such that
fa ( x ) = fa( y ) , then
a ? x = a? y .
x = y ?
fa is one – to – one .
Exercise .
Let FG = { fa | a ? G } be set , show that ( F G , o ) is group .
Proof .
Since G ? ? , then there exist a? G , then fa ? FG .
Hence FG ? ?
If fa ,fb ? FG , then
( fa o fb ) ( x ) = fa ( fb ( x))
= fa ( b ? x )
= a ? ( b ? x )
= ( a ? b ) ? x
= fa? b (x)
So FG is closed under o .
We must prove that
(( fa o f b ) o f c )( x ) = ( fa o ( fb o fc )) (x )
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d1 v

h r

d2
The group of symmetries of a square






Exerciess .
If G = { ( a , b )| a , b ? R and a? o } , then ( G , * ) is
group where (a , b) * (c , d) = ( ac , bc + d )