Residual property (mathematics)

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 h(g)≠e.

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 ϕ:G->H 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.

0.00

(0 votes)

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

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Residual_property_(mathematics)&oldid=232330"