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 [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.