residually $\mathfrak{X}$

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

A group $G$ is said to be residually $\mathfrak{X}$ 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 $\mathfrak{X}$. Equivalently, $G$ is residually $\mathfrak{X}$ if and only if

 $\bigcap_{N\trianglelefteq_{\mathfrak{X}}G}\!\!N=\{1\},$

where $N\trianglelefteq_{\mathfrak{X}}G$ means that $N$ is normal in $G$ and $G/N$ has property $\mathfrak{X}$.

It can be shown that a group is residually $\mathfrak{X}$ if and only if it is isomorphic to a subdirect product of $\mathfrak{X}$-groups. If $\mathfrak{X}$ is a hereditary property (that is, every subgroup (http://planetmath.org/Subgroup) of an $\mathfrak{X}$-group is an $\mathfrak{X}$-group), then a group is residually $\mathfrak{X}$ if and only if it can be embedded in an unrestricted direct product of $\mathfrak{X}$-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.

Title residually $\mathfrak{X}$ ResiduallymathfrakX 2013-03-22 14:53:22 2013-03-22 14:53:22 yark (2760) yark (2760) 15 yark (2760) Definition msc 20E26 AGroupsEmbedsIntoItsProfiniteCompletionIfAndOnlyIfItIsResiduallyFinite residually finite residually nilpotent residually solvable residually soluble