Homomorphisms and the Isomorphism Theorems

A homomorphism is a map between groups that respects their operations. The homomorphisms out of a group are controlled by its normal subgroups, and the precise dictionary between the two is the content of the isomorphism theorems. This page builds on subgroups-cosets.md (normal subgroups and quotients) and supplies the structural language used everywhere downstream.

Homomorphisms

A homomorphism $\varphi :G\to H$ is a map preserving the group operation: $$\varphi (g_1{g}_2)=\varphi (g_1)\varphi (g_2).$$ It automatically preserves the identity and inverses: $$\varphi (e_G)=e_H,\varphi (g^{-1})=\varphi (g{)}^{-1}.$$ A bijective homomorphism is an isomorphism, written $G\cong H$; two isomorphic groups are "the same group with relabelled elements". Special cases:

• Endomorphism — a homomorphism $G\to G$.
• Automorphism — an isomorphism $G\to G$ (see below).
• Monomorphism / epimorphism — injective / surjective homomorphism.

Kernel and image

The kernel is the preimage of the identity: $$\ker \varphi \equiv {\varphi }^{-1}(e_H)=\{g\in G:\varphi (g)=e_H\}\le G,$$ and the image is $\mathrm{im}\varphi =\varphi (G)\le H$.

• $\ker \varphi ◃G$ is always normal (conjugating a kernel element stays in the kernel). Conversely every normal subgroup is a kernel — namely of the quotient map $G\to G/N$, $g\mapsto gN$.
• $\varphi$ is injective iff $\ker \varphi =\{e_G\}$ — only the identity maps to the identity.
• $\mathrm{im}\varphi$ is a subgroup of $H$, not necessarily normal.

Example: the determinant $\det :GL(n,\mathbb{R})\to {\mathbb{R}}^{\times }$ is a homomorphism with kernel $SL(n,\mathbb{R})$. The covering map $SL(2,\mathbb{C})\to S{O}^{+}(1,3)$ has kernel $\{\pm 1\}$ — see topology-covers.md.

The four isomorphism theorems

First (fundamental) isomorphism theorem. For any homomorphism $\varphi :G\to H$, $$G/\ker \varphi \cong \mathrm{im}\varphi .$$

Every homomorphism thus factors as a surjection onto a quotient followed by an injection: $G\twoheadrightarrow G/\ker \varphi \stackrel{\sim }{\to }\mathrm{im}\varphi \hookrightarrow H$. This is the organising principle — "images are quotients".

Second (diamond) isomorphism theorem. If $H\le G$ and $N◃G$, then $HN\le G$, $H\cap N◃H$, and $$H/(H\cap N)\cong HN/N.$$

Third isomorphism theorem. If $N◃K◃G$ with $N◃G$, then $K/N◃G/N$ and $$(G/N)/(K/N)\cong G/K.$$ "Quotients of quotients cancel."

Fourth (correspondence/lattice) theorem. For $N◃G$, the subgroups of $G/N$ correspond bijectively and inclusion-preservingly to the subgroups of $G$ containing $N$, matching normal to normal (stated already in subgroups-cosets.md).

These four are the workhorses behind the Sylow counting arguments and the composition-series machinery.

Automorphisms, the centre, and inner automorphisms

The automorphisms of $G$ form a group $\mathrm{Aut}(G)$ under composition. Each $g\in G$ gives an inner automorphism (conjugation) $$c_g:x\mapsto gx{g}^{-1},$$ and the map $g\mapsto c_g$ is a homomorphism $G\to \mathrm{Aut}(G)$ whose

• kernel is the centre $Z(G)=\{z:zg=gz\forall g\}$, and
• image is the group of inner automorphisms $\mathrm{Inn}(G)$.

By the first isomorphism theorem, $$\mathrm{Inn}(G)\cong G/Z(G).$$ The quotient $\mathrm{Out}(G)=\mathrm{Aut}(G)/\mathrm{Inn}(G)$ is the outer automorphism group. (Famously $\mathrm{Out}(S_n)$ is trivial except for the exceptional $\mathrm{Out}(S_6)\cong {\mathbb{Z}}_2$ — see permutation-groups.md.)

$\mathrm{Inn}(G)◃\mathrm{Aut}(G)$ is normal, because conjugating an inner automorphism by any automorphism yields another inner one. Conjugation also drives the class equation and the whole action-theoretic viewpoint of group-actions.md.

References

• Dummit & Foote, Abstract Algebra, §3.3.
• Rotman, An Introduction to the Theory of Groups, Ch. 2.