= Kernel of a group homomorphism
{title2=$\ker\varphi$}
= Group homomorphism kernel
{synonym}
= Group kernel
{synonym}
The <kernel of a group homomorphism> is the group version of a <kernel (algebra)>. For $\varphi:G\to H$, it is
$$
\ker\varphi=\{g\in G:\varphi(g)=e_H\}.
$$
It is a <normal subgroup> of the domain: for $n$ in the <group kernel>, $\varphi(gng^{-1})=\varphi(g)e_H\varphi(g)^{-1}=e_H$. 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>.
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