Homomorphisms

Definition (Homomorphism)

Let $G$ and $G^{\prime }$ be groups. We say that a mapping $\phi :G\to G^{\prime }$ is a (group) homomorphism if $\phi (xy)=\phi (x)\phi (y)$ for all $x,y\in G$.

Definition (Mono-, Epi- and Isomorphisms)

A group homomorphism is called monomorphism if it is injective, epimorphism if it is surjective, and isomorphism if it is both, i.e. bijective.

Definition (Endo- and Automorphisms)

A homomorphism from a group to itself is called an endomorphism. An isomorphism from a group to itself is called an automorphism.

Clearly, an automorphism is the same as a bijective endomorphism.

isomorphic

Cayley’s Theorem

Every group is isomorphic to a subgroup of the automorphism group of some set.

Definition (Kernel)

The kernel of a group homomorphism $\phi :G\to G^{\prime }$ is defined to be the set of elements of $G$ that are mapped to the neutral element of $G^{\prime }$, that is the set

$$\ker \phi =\{x\in G:\phi (x)=e^{\prime }\}$$

If $\phi :G\to G^{\prime }$ is a group homomorphism, then $\phi$ is a monomorphism if and only if $\ker \phi =\{e\}$.