The Third Isomorphism Theorem
- Suppose
is a normal subgroup of a group and is the usual projection homomorphism, defined by . Prove that if is any homomorphism with , then there exists a unique homomorphism such that . (You must explicitly define , show it is well defined, show , and show that is uniquely determined.) - Prove the:
Third Isomorphism Theorem. Ifwith , then .
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\begin{enumerate}[label=\alph*)]
\item Suppose $N$ is a normal subgroup of a group $G$ and $\pi_N:G\to G/N$ is the usual projection homomorphism, defined by $\pi_N(g)=gN$. Prove that if $\phi:G\to H$ is any homomorphism with $N\leq \ker(\phi)$, then there exists a unique homomorphism $\psi:G/N\to H$ such that $\phi = \psi\circ \pi_N$. (You must explicitly define $\psi$, show it is well defined, show $\phi=\psi\circ\pi_N$, and show that $\psi$ is uniquely determined.)
\item Prove the:
\medskip
{\bfseries Third Isomorphism Theorem.} If $M, N\unlhd G$ with $N\leq M$, then $(G/N)/(M/N)\cong G/M$.
\end{enumerate}