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The first isomorphism theorem

The connection between kernels and normal subgroups induces a connection between quotients and images.


\begin{thm}[The first isomorphism theorem]
If $f\st G\longrightarrow H$\ is a ho...
...phism then
\begin{displaymath}
G/\ker(f)\cong \im(f).
\end{displaymath}\end{thm}


\begin{proof}
For brevity denote $\ker(f)$\ by $N$, and $\im(f)$\ by $K$.
Define...
...displaymath}This completes the proof that $\phi$\ is an isomorphism.
\end{proof}

The importance of the first isomorphism theorem is that one may consider quotients without working with cosets.


\begin{ex}
% latex2html id marker 3556The (necessarily normal) subgroup $n{\ma...
...bb{Z}}={\mathbb{Z}}/\ker(f)\cong\im(f)={\mathbb{Z}}_n.
\end{displaymath}\end{ex}



Edmund F Robertson

11 September 2006