# 2019-14 Residual finite groups

A group $$G$$ is called residually finite if for any nontrivial element $$g$$ of $$G$$, there exists a finite group $$K$$ and a surjective homomorphism $$\rho: G \to K$$ such that $$\rho(g)$$ is a nontrivial element of $$K$$.

Suppose $$G$$ is a finitely generated residually finite group. Show that any surjective homomorphism from $$G$$ to itself is an isomorphism.

GD Star Rating