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 $h(g)\neq e$ .

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 $\phi \colon G\to H$ 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.

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