Homomorphism

$Group homomorphism$

f

is a group homomorphism between

G, G^{'}

\forall x, y \in G, f (x \cdot y) = f (x) ⋄ f (y)

f (e) = e^{'} f (x^{- 1}) = f (x)^{- 1} f (x^{n}) = f (x)^{n}

f

is a Group isomorphism between

G, G^{'}

:

f

is a group homomorphism between

G, G^{'}

and bijective

f

is a group isomophism between

G, G^{'}

\Leftrightarrow f^{- 1}

is a group isomophism between

G, G^{'}

f

is

G, G^{'}

injective group isomorphism

\Leftrightarrow (f (x) = e^{'} \Rightarrow x = e)

G

and

G^{'}

are group isomorphic:

G ≅ G^{'}

:

Group isomophic between

G, G^{'}

exists

g

is a group automorphism of

G

：

g

is a group isomorphism

Automorphic group of

G

:

(A u t (G), \cdot)

ker f := {x \in G ∣ f (x) = e^{'}}

I m f := {f (x) \in G^{'} ∣ x \in G}

ker f

is a normal subgroup of

G

,

I m f

is a subgroup of

G^{'}

G / Ker f ≅ I m f

N

is normal subspace of

N H

H / H \cap N ≅ H N / N

N / K

is a normal subgroup of

G / K

(G / K) / (N / K) ≅ G / N

f^{- 1} (H^{'}) is subgroup of G f^{- 1} (N^{'}) is normal subgroup of G

Canonical group homomorphism between

G, G / N

π : G \to G / N; x \mapsto x N

π

is a surjective group homomorphism and

ker π = N