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In the [[mathematics|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 [[Group homomorphism|homomorphism]] ''h'' from ''G'' to a group with property ''X'' such that <math>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 <math>\phi\colon G \to H</math> where ''H'' is a group with property ''X''.

==Examples==
Important examples include:
* [[Residually finite group|Residually finite]]
* Residually [[nilpotent group|nilpotent]]
* Residually [[solvable group|solvable]]
* Residually [[free group|free]]

==References==
* {{cite book | title=The theory of groups | author=Marshall Hall Jr | authorlink=Marshall Hall (mathematician) | location=New York | publisher=Macmillan | year=1959 | page=16 }}

[[Category:Infinite group theory]]
[[Category:Properties of groups]]


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