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 $x\in G\backslash \{1\}$ 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 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.

Title	residually $𝔛$
Canonical name	ResiduallymathfrakX
Date of creation	2013-03-22 14:53:22
Last modified on	2013-03-22 14:53:22
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	15
Author	yark (2760)
Entry type	Definition
Classification	msc 20E26
Related topic	AGroupsEmbedsIntoItsProfiniteCompletionIfAndOnlyIfItIsResiduallyFinite
Defines	residually finite
Defines	residually nilpotent
Defines	residually solvable
Defines	residually soluble