= Finite residual contains surjective endomorphism kernels
{title2=$\ker\theta\subseteq R_G$}
Let $G$ be a <finitely generated group> and $\theta:G\to G$ a surjective <endomorphism>. Pullback by $\theta$ permutes the finite set of normal subgroups of any prescribed finite index. For each such $N$, some iterate satisfies $\theta^{-m}(N)=N$, so $\ker\theta\subseteq\ker\theta^m\subseteq N$. Intersecting proves the inclusion. Consequently every finitely generated <residually finite group> is a <Hopfian group>.
Back to article page
