# 準同型写像の核

$(G, * )$ and $(H, \star )$ : Group
$e_H$ : the identity element of H
$\phi$: homomorphism $G \to H$ $$\label{def_kernel_hom} \rm{Ker}\phi := \{g \in G| \phi(g) = e_H \}$$ then, satisfies following conditions:
$$\label{kernel_is_nomal} \rm{Ker}\phi \vartriangleleft G$$
(Proof \ref{kernel_is_nomal})
$$\forall x,y\in\rm{Ker}\phi, \phi(xy^{-1})=\phi(x)\phi(y^{-1})=\phi(x)\phi(y)^{-1}=e_H e_H^{-1}=e_H$$ thus, $xy^{-1}\in\rm{Ker}\phi,\rm{Ker}\phi \le G$.
Also, $$\forall g\in G,k\in\rm{Ker}\phi, \phi(gkg^{-1})=\phi(g)\phi(k)\phi(g^{-1}) =\phi(g)e_H\phi(g)^{-1}=e_H e_H e_H^{-1}=e_H$$.