Homomorphism

A homomorphism is a map $\theta :G\to H$ between two groups that preserves the group operation:

$$\boxed{\theta (g_1{g}_2)=\theta (g_1)\theta (g_2)\text{for all }{g}_1,g_2\in G,}$$

where the multiplication on the left is in $G$ and on the right is in $H$.

Basic Properties

For any homomorphism $\theta :G\to H$:

1. $\theta (e_G)=e_H$ - identity goes to identity.
2. $\theta (g^{-1})=\theta (g{)}^{-1}$ - inverses go to inverses.
3. $\theta (g^n)=\theta (g{)}^n$ for all $n\in \mathbb{Z}$.

Each follows from the defining property by short manipulations.

Associated Subgroups

• The Image $\mathrm{Im}(\theta )=\{\theta (g):g\in G\}\le H$.
• The Kernel $\ker (\theta )=\{g\in G:\theta (g)=e_H\}⊴G$ (always normal).

The kernel measures how far from injective $\theta$ is: $\theta$ is injective iff $\ker (\theta )=\{e_G\}$.

Examples

• $\det :{\mathrm{GL}}_n(\mathbb{R})\to {\mathbb{R}}^{\ast }$ - determinant is a homomorphism, kernel ${\mathrm{SL}}_n(\mathbb{R})$.
• $\sigma :S_n\to \{\pm 1\}$ - Signature, kernel $A_n$.
• $\exp :(\mathbb{R},+)\to ({\mathbb{R}}_{>0},\cdot )$ - exponential is an isomorphism.
• For $H⊴G$, the quotient map $G\to G/H$, $g\mapsto gH$, is a surjective homomorphism with kernel $H$.

Bijective Homomorphisms

A bijective homomorphism is an Isomorphism. The two groups are then “the same” up to relabelling.

First Isomorphism Theorem

For any homomorphism $\theta :G\to H$:

$$G/\ker (\theta )\cong \mathrm{Im}(\theta ).$$

The factorisation through the kernel gives a canonical “abstract identification” of source and image - see Isomorphism theorems.