In the
mathematical field of
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
, 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
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
''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
In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related objects, the precise gluing process being specified by morphisms between the objects. Thus, inverse limits ca ...
consisting of all morphisms
from ''G'' to some group ''H'' with property ''X''.
Examples
Important examples include:
*
Residually finite {{unsourced, date=September 2022
In the mathematical field of group theory, a group ''G'' is residually finite or finitely approximable if for every element ''g'' that is not the identity in ''G'' there is a homomorphism ''h'' from ''G'' to a fini ...
* Residually
nilpotent
In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0.
The term was introduced by Benjamin Peirce in the context of his work on the class ...
* 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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