# Residual property (mathematics)

From HandWiki

In the mathematical field of group theory, a group is **residually X** (where

*X*is some property of groups) if it "can be recovered from groups with property

*X*". Formally, a group

*G*is residually

*X*if for every non-trivial element

*g*there is a homomorphism

*h*from

*G*to a group with property

*X*such that [math]\displaystyle{ h(g)\neq e }[/math].

More categorically, a group is residually *X* if it embeds into its pro-*X* completion (see profinite group, pro-p group), that is, the inverse limit of the inverse system consisting of all morphisms [math]\displaystyle{ \phi\colon G \to H }[/math] from *G* to some group *H* with property *X*.

## Examples

Important examples include:

- Residually finite
- Residually nilpotent
- Residually solvable
- Residually free

## References

- Marshall Hall Jr (1959).
*The theory of groups*. New York: Macmillan. p. 16.

Original source: https://en.wikipedia.org/wiki/Residual property (mathematics).
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