Let $\X$ be a property of groups, assumed to be an isomorphic invariant (that is, if a group $G$ has property $\X$ then every group isomorphic to $G$ also has property $\X$ . We shall sometimes refer to groups with property $\X$ as $\X$ groups.

A group $G$ is said to be <</SPAN>#59#>residually $\X$ if for every $x\in G\backslash\{1\}$ there is a normal subgroup $N$ of $G$ such that $x\notin N$ and $G/N$ has property $\X$ Equivalently, $G$ is residually $\X$ if and only if $$ \bigcap_{N\normal_\X G}\!\!N=\{1\}, $$ where $N\normal_\X G$ means that $N$ is normal in $G$ and $G/N$ has property $\X$

It can be shown that a group is residually $\X$ if and only if it is isomorphic to a subdirect product of $\X$ groups. If $\X$ is a hereditary property (that is, every subgroup of an $\X$ group is an $\X$ group), then a group is residually $\X$ if and only if it can be embedded in an unrestricted direct product of $\X$ groups.

It can be shown that a group $G$ is residually solvable if and only if the intersection of the derived series of $G$ is trivial (see transfinite derived series). Similarly, a group $G$ is residually nilpotent if and only if the intersection of the lower central series of $G$ is trivial.