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 . 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  from G to some group H with property X. Examples
Important examples include:
 Residually finite
 Residually nilpotent
 Residually solvable
 Residually free

References
 

Infinite group theory
Properties of groups