5.Homomorphism and isomorphism of groups

definition

Let $<G,\ast >$ and $<H,\circ >$ are groups, mapping$f:G\to H$, and

$\mathrm{\forall }a,b\in G$, have $f(a\ast b)=f(a)\circ f(b)$ (Homomorphic equation),

then $f$ is group homomorphic mapping from $<G,\ast >$ to $<H,\circ >$​.

When $f$​ is injective, surjective, and bijective, respectively, it is called a single group homomorphism, a full group homomorphism, and a group isomorphism, respectively.

Property

Let $f$ be group homomorphism from $<G,\ast >$ to $<H,\circ >$, then:

(1) If $e$ is identify element of group $G$, then $f(e)$ is identify element of group $H$.

(2) For $\mathrm{\forall }a\in G$, have $f(a^{-1})=(f(a){)}^{-1}$.

(3) The homomorphic image $f(G)$ is a subgroup of $H$, $f(G)\le H$​.

Homomorphic kernel

Let $<G,\ast >$ and $<H,\circ >$ be groups, $f:G\to H$ is a group homomorphism, then let:

$$K=\{a|a\in Gandf(a)=e^{\prime }\}$$

which $e^{\prime }$ is identity element of $H$, then $K$ is Homomorphic kernel of $f:G\to H$, denoted as $Kerf$​.

$Kerf$ is a subgroup of $G$.