The Third Isomorphism Theorem

#group_theory

(a) Suppose $N$ is a normal subgroup of a group $G$ and $π_{N} : G \to G / N$ is the usual projection homomorphism, defined by $π_{N} (g) = g N$ . Prove that if $ϕ : G \to H$ is any homomorphism with $N \leq \ker (ϕ)$ , then there exists a unique homomorphism $ψ : G / N \to H$ such that $ϕ = ψ \circ π_{N}$ . (You must explicitly define $ψ$ , show it is well defined, show $ϕ = ψ \circ π_{N}$ , and show that $ψ$ is uniquely determined.)
(b) Prove the:
Third Isomorphism Theorem. If $M, N ⊴ G$ with $N \leq M$ , then $(G / N) / (M / N) ≅ G / M$ .

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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}