| Version 8 |
Version 7 |
| Let $\rho :G\to K$ be a group homomorphism. The preimage of the |
Let $\rho :G\to K$ be a group homomorphism. The set of |
| codomain identity element $e_K\in K$ forms a subgroup of the domain |
all elements in $G$ that map to $e_K$(the identity element) is called the kernel of $\rho$. The kernel is denoted:\begin{center}$\operatorname{ker}(\rho)= \{ s |
| $G$, called the \emph{kernel} of the homomorphism; |
\in G|\rho (s)=e_K\} $\end{center} The kernel is a also a normal subgroup. |
| $$\operatorname{ker}(\rho)= \{ s \in G\mid\rho (s)=e_K\} $$ |
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| The kernel is a normal subgroup. It is the trivial subgroup if and |
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| only if $\rho$ is a monomorphism. |
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