The Isomorphism Theorems

A homomorphism compresses a group onto its image, and its kernel records what gets crushed to the identity. The isomorphism theorems make that compression exact: the image is a copy of the quotient by the kernel, and every quotient arises this way. Three further theorems handle quotients assembled from two subgroups, quotients of quotients, and the correspondence of subgroup lattices.

The first isomorphism theorem

The kernel is normal because it is a kernel. The isomorphism sends the coset $g\ker \phi$ to $\phi (g)$; this is well defined and injective precisely because two elements share a coset exactly when they have the same image, and it is surjective onto $\phi (G)$ by construction.¹ Equivalently, a homomorphism factors into a projection followed by an isomorphism.

The second part is a counting law: the number of cosets of the kernel equals the size of the image. For linear maps of vector spaces it reappears as $\dim V=\mathrm{rank}\phi +\mathrm{nullity}\phi$, the rank-nullity theorem.²

The other three theorems are consequences of the first applied to well-chosen homomorphisms.

The second (diamond) isomorphism theorem

Since $A$ normalizes $B$, the product $AB$ is a subgroup and $B$ is normal in it. Define $\phi :A\to AB/B$ by $\phi (a)=aB$; it is a surjective homomorphism with kernel $\{a\in A\mid aB=B\}=A\cap B$, and the first theorem finishes it.⁵ The name comes from the shape of the four subgroups in the lattice: $AB$ on top, $A$ and $B$ on the sides, $A\cap B$ at the bottom, with the two slanted edges carrying isomorphic quotients.

This recovers the product-order formula $|AB|=|A||B|/|A\cap B|$ in the special case $A\le N_G(B)$, now as an isomorphism rather than a count. Two cautions attach to the picture. The unmarked edges of the diamond are not isomorphic to each other in general: $AB/A$ need not even be a group, since $A$ need not be normal in $AB$; only the index equation $[AB:A]=[B:A\cap B]$ survives on that side. And the hypothesis $A\le N_G(B)$ is automatic whenever $B⊴G$, which is the usual way the theorem is invoked: for any subgroup $A$ and any normal subgroup $B$, the conclusion holds with no further checking.

The third isomorphism theorem

Define $\phi :G/H\to G/K$ by $gH\mapsto gK$; because $H\le K$, this is well defined, and it is a surjective homomorphism with kernel $K/H$.⁶ The mnemonic is invert and cancel, as with fractions: the two $H$'s cancel. The theorem says taking a quotient of a quotient yields no structure that a single quotient does not already give.

For a numeric instance, take $G=\mathbb{Z}/24\mathbb{Z}$ with $H=⟨\overline{12}⟩$ and $K=⟨\bar{6}⟩$, so $H\le K$ and both are normal (the group is abelian). Then $G/H\cong \mathbb{Z}/12\mathbb{Z}$ from the first-theorem example above, $K/H$ is the order-$2$ subgroup $⟨\bar{6}⟩/⟨\overline{12}⟩$ inside it, and the theorem promises $(G/H)/(K/H)\cong G/K\cong \mathbb{Z}/6\mathbb{Z}$ — the fraction $\frac{24/12}{6/12}$ canceling to $\frac{24}{6}$.

The fourth (lattice) isomorphism theorem

The last theorem is the reason the quotient's lattice can be read off the lattice of $G$: collapse $N$ to a point, and everything above $N$ survives, faithfully.

The correspondence sends each subgroup above $N$ to its image, and each subgroup of $G/N$ back to its complete preimage under the natural projection.⁷ It preserves the entire lattice architecture: containment, index, joins, meets, and normality. The lattice of $G/N$ coincides with the top of the lattice of $G$, the part sitting above $N$.

Defining maps on quotients

The theorems share a mechanism: to build a homomorphism out of $G/N$, build one out of $G$ and check that it sends $N$ to $1$. A homomorphism $\Phi :G\to H$ factors through $N$, inducing a well-defined $\phi :G/N\to H$ with $\phi (gN)=\Phi (g)$, if and only if $N\le \ker \Phi$.⁸ This single criterion is what makes each isomorphism above well defined.

Quotient information also flows back down to $G$. A standard specimen:

If $G/Z(G)=⟨xZ(G)⟩$, every coset is $x^{a}Z(G)$, so every element of $G$ is $x^{a}z$ with $z$ central. Two such elements commute:

since powers of $x$ commute with each other and $z_1,z_2$ commute with everything.⁹ The proposition is a template for the whole method: prove something about the small group $G/N$, and pull the conclusion back through the projection. Its immediate payoff is that any group of order $p^2$ (prime $p$) is abelian — the center is nontrivial by a counting argument met in the class equation, so $G/Z(G)$ has order $1$ or $p$, both cyclic.

With these theorems, a quotient can be identified with a familiar group through its image rather than handled as an abstract set of cosets.