Advanced Abstract Algebra - Maharshi Dayanand University, Rohtak

UNIT-I
15
Definition: Group Homomorphism
Let be a mapping from a group G to a group defined by
: G
a
bg bgbg
f ab = f a f b ∀a, b ∈G.
f is called homomorphism of groups.
Definition: Kernel of a Homomorphism
Let : G
be a group homomorphism and be the identify of
denoted by Ker is defined by
. Then Kernel of
Ker
=
G
homomorphism
Ker
We note that ker G. (It is easy to show that ker f is a subgroup of G)
Θm
g f∆
−
xbg∀ a Ga : ∈ f Gbg
x,
Satisfying
1 ker b g
⊆ ker
−1 f g f f∀ gx ∈
G
−
g f g ef g x f
We show that
= 1
Ge
x
fd G,
−1
∈( g xgf
() ix
= ) e = ∀d xf
∈( xi ker ) ( bgbg Θf
f,
( x)
∈Gdand
ie
isbg, the identify Θ ∈ kerof
G )
Let x be any element of ker
.
Then
−1 −1
d i bg d i b
f g f g = f g g Θ f ishomomorphism
bg
= f e
= e
( Any homomorphism of groups carries identity of G to identity of
Explanation:
g
)
=
=
f
f
( xe)
( x) f ( e) in G
So by cancellation property in
, we have
=
(e).)
Hence
g −1
xg ∈ ker f ∀g
∈G,
∀ x ∈ker
f ker f∆G