HOMOMORPHISMS

HOMOMORPHISMS
Definition 3 – 1 .
Let ( G , * ) and ( G/ , ?) be two groups and ?? a function from . G into G/. Then ?? is said to be a HOMOMORPHISM from ( G ,* ) . into ( G/ , ? ) if and only if
?? ( a * b ) = ?? (a) ???(b) for every a , b ?G.
Example 1.
Let (G , * ) be a group , define the function ?? : G ? G by . f(a) = a for all a ?G ,
then ?? is homomorphism .
Solution
If a , b ? G , then
??( a * b ) = a * b = ?? ( a ) * ?? ( b ) .
Then ?? is homomorphism.
?? is said identity map on G , and denoted by iG .
Example 2.
If ?? : Z? Z n define as
??( n) = [n] , then ?? is homomorphism .
Example 3 .
Show that the mapping defined by ??(n) = in where i2 = -1and
n?Z is a homomorphism from (Z, +) onto { 1,-1,i,-i} .
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Note
We shall lable the set of all homomorphisms from the group . ( G , * ) into itself by hom G .
Theorem 3 -1 .
( hom G , o ) is semigroup with identity .
Proof.
If ?? , ??? hom G , then
( ?? o??) ( a* b ) = ?? (?? ( a * b ))
=??(?? ( a ) * ?? ( b) )
=?? (?? (a) ) * ??(??(b))
= (?? o??) (a) * (?? o ?? ) ( b) .
hom G is closed .
The composition of functiones is associative ?
So (hom G , o ) is semigroup .
iG is the identity ?
Theorem 3 -2 .
If ?? is a homomorphism from the group ( G , *) into the . group ( G/ , ? ) then
1) ?? (e) = e/where e/ is the identity of G/ .
?? (a-1) =?? (a)-1 for each a ? G .
Proof .
1) ?? (a) ? e/ = ??(a) = ?? (a * e) = ?? (a) ??? (e)?
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Hence ??(e)= e/ ?
2)??(a) ??? (a-1) = ?? (a * a-1 ) = ?? (e) = e/
Similarly ??(a-1) ??? (a) = e/.
Then ?? (a-1) = ?? (a)-1 .
Theorem 3 -3 .
Let ?? is a homomorphism from the group ( G , *) into . the group ( G / , ? ) .
For each subgroup (H , * ) , then (??(H) , ? ) is subgroup
of ( G / , ? ) .
For each subgroup ( H/ , ? ) from ( G / , ? ) , then
(??-1(H/) , *) is subgroup of (G , * ) .
Proof .
1) ??(H)?? ?
Let x , y ? ??(H) , then ? h1 , h2 ?H?
x = ??(h1) and y = ?? (h2)
x ? y-1 = ??(h1) ? ??(h2)-1
= ??(h1) ? ??(h2-1)
= ??( h1 * h2-1)
Since ( h1 * h2-1) ? H , then ?? ( h1 * h2-1) ? ??(H)
So x ? y-1 ? ??(H).
Hence (??(H) , ? ) is subgroup of ( G / , ? ) .