residually $𝔛$

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 residually $\mathrm{X}$if for every $x\in G\backslash \backslash \{1\}$ 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

$$\bigcap _{N{\mathrm{⊴}}_{𝔛}G}N=\{1\},$$

where $N{\mathrm{⊴}}_{𝔛}G$ 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 (http://planetmath.org/Subgroup) of an $𝔛$-group is an $𝔛$-group),then a group is residually $𝔛$ if and only ifit 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 ifthe intersection of the derived series of $G$ is trivial(see transfinite derived series).Similarly, a group $G$ is residually nilpotent if and only ifthe intersection of the lower central series of $G$ is trivial.