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 $a\ne b$ 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 $f\left(a{b}^{-1}\right)=e_H$; in other words, $a{b}^{-1}\in \ker f$, and ker f would not be the singleton. Conversely, distinct elements of the kernel violate injectivity directly: if there would exist an element $g\ne e_G\in \ker f$, then $f(g)=f(e_G)=e_H$, thus f would not be injective.