Residual property (mathematics)

In the mathematical field of group theory, a group is residually "X" (where "X" is some property of groups) if it "can be recovered from groups with property "X"".
Formally, a group "G" is residually "X" if for every non-trivial element "g" there is a homomorphism "h" from "G" to a group with property "X" such that formula_1.
More categorically, a group is residually "X" if it embeds into its pro-"X" completion (see profinite group, pro-p group), that is, the inverse limit of the inverse system consisting of all morphisms formula_2 from "G" to some group "H" with property "X".
Examples.
Important examples include: