PlanetMath: residually $\mathfrak{X}$

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (198)
Orphanage
Unclass'd
Unproven (490)
Corrections (7)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

residually $\mathfrak{X}$

(Definition)

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

A group is said to be residually $\mathfrak{X}$ if for every $x\in G\backslash\{1\}$ there is a normal subgroup of such that $x\notin N$ and has property $\mathfrak{X}$ . Equivalently, is residually $\mathfrak{X}$ if and only if

$\begin{displaymath} \bigcap_{N\trianglelefteq _\mathfrak{X}G}\!\!N=\{1\}, \end{displaymath}$

where $N\trianglelefteq _\mathfrak{X}G$ means that

is normal in

and

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 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 is residually solvable if and only if the intersection of the derived series of is trivial (see transfinite derived series). Similarly, a group is residually nilpotent if and only if the intersection of the lower central series of is trivial.