The isomorphism theorems explain why kernels, images, quotients, and subobjects fit together so rigidly. They are the main grammar of algebraic comparison.
The first isomorphism theorem
Theorem. If \(\varphi:G\to H\) is a group homomorphism, then \[ G/\ker\varphi\cong \operatorname{im}\varphi, \qquad g\ker\varphi\mapsto \varphi(g). \]
Proof. The proposed map is well-defined because \(g\ker\varphi=h\ker\varphi\) iff \(h^{-1}g\in\ker\varphi\), iff \(\varphi(g)=\varphi(h)\). It is a surjective homomorphism onto \(\operatorname{im}\varphi\), and its kernel is the identity coset.
Example. The sign map \(\operatorname{sgn}:S_n\to\{\pm1\}\) has kernel \(A_n\). Hence \(S_n/A_n\cong C_2\). This turns parity of permutations into a quotient group.
The correspondence theorem
Theorem. Let \(N\trianglelefteq G\). Subgroups of \(G/N\) correspond bijectively to subgroups \(K\le G\) containing \(N\), by \[ K\longmapsto K/N, \qquad \overline K\longmapsto \pi^{-1}(\overline K). \] Normal subgroups correspond to normal subgroups.
Remark. This theorem says quotienting does not destroy subgroup information above the kernel; it repackages it inside \(G/N\).
The second isomorphism theorem
Theorem. If \(A\le G\) and \(N\trianglelefteq G\), then \(AN\le G\), \(A\cap N\trianglelefteq A\), and \[ A/(A\cap N)\cong AN/N. \]
Proof. Restrict the quotient map \(\pi:G\to G/N\) to \(A\). Its image is \(AN/N\), and its kernel is \(A\cap N\). Apply the first isomorphism theorem.
The third isomorphism theorem
Theorem. If \(N\trianglelefteq K\trianglelefteq G\), then \(K/N\trianglelefteq G/N\) and \[ (G/N)/(K/N)\cong G/K. \]
Proof. Define \(G/N\to G/K\) by \(gN\mapsto gK\). This is well-defined because \(N\subset K\). Its kernel is \(K/N\), and the first isomorphism theorem gives the result.
Exact sequences as compressed isomorphism data
Definition. A sequence \(1\to A\xrightarrow{i}B\xrightarrow{p}C\to1\) is exact if \(i\) is injective, \(p\) is surjective, and \(\operatorname{im}i=\ker p\). Then \(C\cong B/i(A)\).
Example. The sequence \[ 1\to A_n\to S_n\xrightarrow{\operatorname{sgn}} C_2\to1 \] is exact. It records \(S_n/A_n\cong C_2\) and packages the kernel-image relation in one line.
A warning about canonical isomorphisms
Remark. The isomorphism theorems usually produce canonical maps. This is stronger than merely saying two groups have the same size or abstract isomorphism type: the map is built from the original homomorphism or quotient map.