# 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.

Major Section:
Reference
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## Mathematics Subject Classification

20A05 Axiomatics and elementary properties

## 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.

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## Info

Owner: yark
Added: 2001-11-08 - 21:24
Author(s): yark

## Versions

(v27) by unlord 2013-05-17
(v26) by yark 2013-05-17
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