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Homegroup homomorphism

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# group homomorphism

Let $(G,\ast)$ and $(K,\star)$ be two groups. A *group homomorphism* is a
function^{} $\phi\colon G\to K$ such that
$\phi(s\ast t)=\phi(s)\star\phi(t)$ for all $s,t\in G$.

A composition of group homomorphisms is again a homomorphism^{}.

Let $\phi\colon G\to K$ a group homomorphism.
Then the kernel of $\phi$ is a normal subgroup of $G$,
and the image of $\phi$ is a subgroup^{} of $K$.
Also, $\phi(g^{n})=\phi(g)^{n}$ for all $g\in G$ and for all $n\in\mathbb{Z}$.
In particular,
taking $n=-1$ we have $\phi(g^{{-1}})=\phi(g)^{{-1}}$ for all $g\in G$,
and taking $n=0$ we have $\phi(1_{G})=1_{K}$,
where $1_{G}$ and $1_{K}$ are the identity elements of $G$ and $K$,
respectively.

Some special homomorphisms have special names.
If the homomorphism $\phi\colon G\to K$ is injective^{},
we say that $\phi$ is a *monomorphism ^{}*,
and if $\phi$ is surjective

^{}we call it an

*epimorphism*. When $\phi$ is both injective and surjective (that is, bijective) we call it an

*isomorphism*. In the latter case we also say that $G$ and $K$ are

^{}*isomorphic*, meaning they are basically the same group (have the same structure). A homomorphism from $G$ on itself is called an

*endomorphism*, and if it is bijective then it is called an

*automorphism*.

## Mathematics Subject Classification

20A05*no label found*

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## Comments

## Re: proofs

By all means, please add the proofs. While things like biographies

of politicians (unless those politicians also happened to be

mathematicians) are off topic, proofs of mathematical statements,

however humble are definitely on topic. Even if this isn't the

most profound theorem, it is definitely worth having for completeness

and could be of use to someone new to algebra or needing a formal

proof of this fact which most mathematicians take for granted.

## Re: proofs

Look at the "latest addtioins" column on the right of the page.