Finitely generated residually finite group
From Groupprops
This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: finitely generated group and residually finite group
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This is a variation of finite group|Find other variations of finite group |
Contents
Definition
A finitely generated residually finite group is a group satisfying the following equivalent conditions:
- It is both finitely generated (i.e., it has a finite generating set) and residually finite (i.e., for every non-identity element, there exists a normal subgroup of finite index not containing that element).
- It is finitely generated, and for every non-identity element, there is a subgroup of finite index not containing that element.
- It is finitely generated and, for any non-identity element, there is a characteristic subgroup of finite index not containing that element.
Equivalence of definitions
- The equivalence of definitions (1) and (2) follows from Poincare's theorem, which in particular asserts that any subgroup of finite index contains a normal subgroup of finite index. Note that this equivalence simply relies on the equivalence of two formulations of the definition of residually finite group, and does not directly involve finite generation.
- The equivalence of these with definition (3) uses the fact that finitely generated implies every subgroup of finite index contains a characteristic subgroup of finite index.
Relation with other properties
Stronger properties
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| finitely generated group | has a finite generating set | finitely generated not implies residually finite | Finitely generated Hopfian group|FULL LIST, MORE INFO | |
| residually finite group | every non-identity element is outside a normal subgroup of finite index | |FULL LIST, MORE INFO | ||
| Hopfian group | any surjective endomorphism is an automorphism | finitely generated and residually finite implies Hopfian | Finitely generated Hopfian group|FULL LIST, MORE INFO | |
| finitely generated Hopfian group | finitely generated and Hopfian | finitely generated and residually finite implies Hopfian | |FULL LIST, MORE INFO |
