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
.
More
categorically, a group is residually ''X'' if it embeds into its pro-''X'' completion (see
profinite group In mathematics, a profinite group is a topological group that is in a certain sense assembled from a system of finite groups.
The idea of using a profinite group is to provide a "uniform", or "synoptic", view of an entire system of finite groups. ...
,
pro-p group
In mathematics, a pro-''p'' group (for some prime number ''p'') is a profinite group G such that for any open normal subgroup N\triangleleft G the quotient group G/N is a ''p''-group. Note that, as profinite groups are compact, the open subgro ...
), that is, the
inverse limit of the
inverse system consisting of all morphisms
from ''G'' to some group ''H'' with property ''X''.
Examples
Important examples include:
*
Residually finite
* Residually
nilpotent
* Residually
solvable
* Residually
free
Free may refer to:
Concept
* Freedom, having the ability to do something, without having to obey anyone/anything
* Freethought, a position that beliefs should be formed only on the basis of logic, reason, and empiricism
* Emancipate, to procur ...
References
*
Infinite group theory
Properties of groups
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