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Section 1.15 Fourth isomorphism theorem
Theorem 1.15.1 . Fourth isomorphism theorem.
Let \(\phi\colon G\rightarrow H\) be a surjective homomorphism, and let \(N=\ker\phi\text{.}\) Define
\begin{align*}
\mathcal{S}_{G,N} \amp =\{K\subseteq G \colon N\leq K\leq G\}\\
\mathcal{S}_H \amp =\{K\subseteq H\mid K\leq H\}\text{.}
\end{align*}
In other words, \(\mathcal{S}_{G,N}\) is the set of all subgroups of \(G\) containing \(N\text{,}\) and \(\mathcal{S}_H\) is the set of all subgroups of \(H\text{.}\)
There is a bijective correspondence between
\(\mathcal{S}_{G,N}\) and
\(\mathcal{S}_H\) given as follows:
For \(K,K'\in \mathcal{S}_{G,N}\text{,}\) we have
\begin{align}
K \text{ normal } \amp \iff \phi(K) \text{ normal} \tag{1.15.1}\\
K\subseteq K' \amp \iff \phi(K)\subseteq \phi(K')\tag{1.15.2}\\
\phi(K\cap K') \amp = \phi(K)\cap \phi(K')\tag{1.15.3}\\
\phi(\angvec{K,K'}) \amp =\angvec{\phi(K),\phi(K')}\tag{1.15.4}
\end{align}
Proof.
Corollary 1.15.2 . Fourth isomorphism theorem: quotients.
Let
\(N\) be a normal subgroup of
\(G\text{.}\) The subgroups of
\(G/N\) are in bijective correspondence with the subgroups of
\(G\) containing
\(N\text{.}\) Furthermore, this correspondence preserves normality, inclusion, intersection, generated subgroups, etc..
