|
|
|
Viewing Version
17
of
'group homomorphism'
|
[ view 'group homomorphism'
|
back to history
]
| Title of object: |
group homomorphism |
| Canonical Name: |
GroupHomomorphism |
| Type: |
Definition |
| Created on: |
2001-11-08 16:24:02 |
| Modified on: |
2005-07-23 12:08:51 |
| Classification: |
msc:20A05 |
| Defines: |
epimorphism, monomorphism, automorphism, endomorphism, homomorphic, isomorphic |
Revision comment (for changes between this and next version):
| Changes for correction #9474 ('emphasis on defined terms'). |
Preamble:
Content:
Let $(G,*_g)$ and $(K,*_k)$ be two groups. A group homomorphism is a
function $\phi\colon G \to K$ such that $\phi (s *_g t) = \phi(s) *_k
\phi(t)$ for all $s,t \in G$.
The composition of group homomorphisms is again an homomorphism.
Let $\phi\colon G\to K$ a group homomorphism. Then
\begin{itemize}
\item $\phi (e_g) = e_k$ where $e_g$ and $e_k$ are the respective identity elements for $G$ and $K$.
\item $\phi (g)^{-1} = \phi (g^{-1})$ for all $g \in G$
\item $\phi(g)^z=\phi(g^z)$ for all $g\in G$ and for all $z \in\mathbbmss{Z}$
\end{itemize}
The kernel of $\phi$ is a subgroup of $G$ and its image is a subgroup of $K$.
Some special homomorphisms have special names.
If $\phi\colon G\to K$ is injective, we say that $\phi$ is an monomorphism, and if $\phi$ is onto 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). An homomorphism from $G$ on itself is called an endomorphism, and if it is bijective, then is called an automorphism. |
|
|
|
|
|
