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