Residually finite group

Summary

A group is residually finite if every element other than the identity survives in some finite quotient. So the group can be studied through its finite images, which separate all its elements. Free groups, finitely generated abelian groups and finitely generated subgroups of $GL(n, \C)$ are residually finite. A finitely generated residually finite group is Hopfian, and a finitely presented residually finite group has solvable word problem.