Examples of homomorphism, endomorphism and automorphism

Question

If $(G, *)$ and $(H, +)$ are both groupoids and if $f: G \to H$ is a surjective function such that $(\forall x,y \in G) f(x*y) = f(x) + f(y)$ than $f$ is called a homomorphism.

If $G=H$ the function is called endomorphism and if $f$ is bijective than it's called automorphism.

For automorphism an example would be the permutations of a set.

For homomorphism an example i found if a function $f(z) = |z|$ with property $f: C_0 \to R_0$ where $C_0,R_0$ are the usual complex and real number sets but without the element $0$.

I can't find an example of endomorphism. I guess the permutations of a set are endomorphism since the function is bijective so in itself it is surjective but i need a pure example. Could you provide one or two :)?

Answer 1

The function $f(x)=2x$ is an endomorphism of $\mathbb Z$; it is obviously non surjective.

Answer 2

Good examples are always:

Any $A\in \text{GL}_n(K)$ is a Automorphism from $(K^n,+)\longrightarrow (K^n,+)$.

Additionally every Automorphism ist also an Endomorphism.

Second good example for an Automorphism is:

The complex conjugate $(\mathbb{C},+)\longrightarrow(\mathbb{C},+): z\mapsto \bar{z}$

As a third example for a surjective but not injective Homomorphism:

$\pi:(\mathbb{Z},+)\longrightarrow (\mathbb{Z}/n\mathbb{Z},+): z\mapsto z+n\mathbb{Z}$

Lastly there is:

$\exp: (\mathbb{R},+)\longrightarrow (\mathbb{R}\setminus\lbrace 0\rbrace,\cdot): x\mapsto e^x$

which is injective but not surjective