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.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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