# Kernel (algebra)

The kernel of a group homomorphism from G to H is the subset of the elements from G that arrive to the Identity element of H.

Mathematically: ${\displaystyle \ker f=\{g\in G:f(g)=e_{H}\}}$. Since a group homomorphism preserves identity elements, the identity element of G must belong to the kernel subset.

## Property

The homomorphism is injective if and only if its kernel is only the identity elements of G.

Proof: If f were not injective, then the non-injective elements can form a distinct element of its kernel: there would exist ${\displaystyle a,b\in G}$ such that ${\displaystyle a\neq b}$ and ${\displaystyle f(a)=f(b)}$. Thus ${\displaystyle f(a)f(b)^{-1}=e_{H}}$. f is a group homomorphism, so inverses and group operations are preserved, giving ${\displaystyle f\left(ab^{-1}\right)=e_{H}}$; in other words, ${\displaystyle ab^{-1}\in \ker f}$, and ker f would not be the singleton. Conversely, distinct elements of the kernel violate injectivity directly: if there would exist an element ${\displaystyle g\neq e_{G}\in \ker f}$, then ${\displaystyle f(g)=f(e_{G})=e_{H}}$, thus f would not be injective.