 Kernel of a group homomorphism

The kernel of a group homomorphism is the group version of a kernel. For $\phi :G\to H$, it isIt is a normal subgroup of the domain: for $n$ in the group kernel, $\phi (gn{g}^{-1})=\phi (g)e_H\phi (g{)}^{-1}=e_H$. The group kernel describes inputs whose effect disappears after mapping. For reduction modulo three from $\mathbb{Z}/6\mathbb{Z}$, it is $\{0,3\}$; for the sign of a permutation on $S_3$, it is $A_3$. A group homomorphism is injective exactly when its group kernel contains only the identity element.