Kernel

For a Homomorphism $\theta :G\to H$, the kernel is

$$\boxed{\ker (\theta )=\{g\in G:\theta (g)=e_H\}\subseteq G.}$$

The set of elements that get sent to the identity. Always a Normal subgroup of $G$.

Why Normal

For any $g\in G$ and $k\in \ker (\theta )$:

$$\theta (g^{-1}kg)=\theta (g{)}^{-1}\theta (k)\theta (g)=\theta (g{)}^{-1}{e}_H\theta (g)=e_H,$$

so $g^{-1}kg\in \ker (\theta )$. Hence $\ker (\theta )⊴G$.

Why It Matters

The kernel measures how far from injective $\theta$ is:

$$\theta \text{ is injective}⟺\ker (\theta )=\{e_G\}.$$

Combined with the First Isomorphism Theorem:

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

the kernel is the unique normal subgroup such that quotienting it out gives an isomorphic copy of the image.

Examples

• $\det :{\mathrm{GL}}_n(\mathbb{R})\to {\mathbb{R}}^{\ast }$: kernel = ${\mathrm{SL}}_n(\mathbb{R})$ (matrices of determinant 1).
• Signature $\sigma :S_n\to \{\pm 1\}$: kernel = $A_n$ (the Alternating group).
• Quotient map $G\to G/N$, $g\mapsto gN$: kernel = $N$.
• $\exp :\mathbb{C}\to {\mathbb{C}}^{\ast }$: kernel = $2\pi i\mathbb{Z}$.

Connection to Group Actions

For a Group action $\lambda :G\to \mathrm{Sym}(X)$, the kernel is the set of elements fixing every point of $X$ - sometimes called the kernel of the action. Distinct from the Stabiliser of a single point.