| Version 13 |
Version 12 |
| Let $(G,*_g)$ and $(K,*_k)$ be two groups. A group homomorphism is a |
Let $(G,*_g)$ and $(K,*_k)$ be two groups. A group homomorphism is a |
| function $\phi :G \to K$ such that $\phi (s *_g t) = \phi(s) *_k |
function $\phi :G \to K$ such that $\phi (s *_g t) = \phi(s) *_k |
| \phi(t)$ for all $s,t \in G$. |
\phi(t)$ for all $s,t \in G$. |
| The composition of group homomorphisms is again an homomorphism. |
The composition of group homomorphisms is again an homomorphism. |
| Let $\phi\colon G\to K$ a group homomorphism. Then |
Let $\phi\colon G\to K$ a group homomorphism. Then |
| \begin{itemize} |
\begin{itemize} |
| \item $\phi (e_g) = e_k$ where $e_g$ and $e_k$ are the respective identity elements for $G$ and $K$. |
\item $\phi (e_g) = e_k$ 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 $\phi (g)^{-1} = \phi (g^{-1})$ for all $g \in G$ |
| \item $\phi(g)^z=\phi(g^z)$ for all $g\in G$ and for all $z \in\mathbbmss{Z}$ |
\item $\phi(g)^z=\phi(g^z)$ for all $g\in G$ and for all $z \in\mathbbmss{Z}$ |
| \end{itemize} |
\end{itemize} |
| The kernel of $\phi$ is a subgroup of $G$ and its image is a subgroup of $K$. |
The kernel of $\phi$ is a subgroup of $G$ and its image is a subgroup of $K$. |
| Some especial homomorphisms have special names. |
Some especial homomorphisms have special names. |
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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.
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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 beijective, then is called an automorphism.
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