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 .
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 from G to some group H with property X.
Examples
[edit]Important examples include:
- Residually finite
- Residually nilpotent
- Residually solvable
- Residually free
References
[edit]- Marshall Hall Jr (1959). The theory of groups. New York: Macmillan. p. 16.