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.

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About Hyungryul

2003.3-2009.8 KAIST, Undergraduate student in Mathematics 2009.8-2014.8 Cornell University, PhD student in Mathematics 2014.9-2017.2 University of Bonn, Postdoc 2017.3-2021.2. KAIST, Assistant Professor 2021.3-Present. KAIST, Associate Professor