Residually Solvable Group

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

*

Infinite group theory
Properties of groups

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