# Residual property (mathematics)

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 $h(g)\neq e$.
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 $\phi\colon G \to H$ where H is a group with property X.