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 ${\displaystyle h(g)\neq e}$.

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 ${\displaystyle \phi \colon G\to H}$ 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.