Finitely generated residually finite group

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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 |

Definition

A finitely generated residually finite group is a group satisfying the following equivalent conditions:

  1. 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).
  2. It is finitely generated, and for every non-identity element, there is a subgroup of finite index not containing that element.
  3. 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

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
finitely generated free group finitely generated and a free group free implies residually finite Finitely presented conjugacy-separable group, Finitely presented residually finite group|FULL LIST, MORE INFO
finitely generated abelian group finitely generated and an abelian group Finitely generated conjugacy-separable group, Finitely presented conjugacy-separable group, Finitely presented residually finite group|FULL LIST, MORE INFO
finitely generated conjugacy-separable group |FULL LIST, MORE INFO
finitely presented residually finite group |FULL LIST, MORE INFO
finitely presented conjugacy-separable group Finitely generated conjugacy-separable group, Finitely presented residually finite group|FULL LIST, MORE INFO
finitely generated profinite group finitely generated and a profinite group |FULL LIST, MORE INFO
finite group underlying set is finite Finitely generated profinite group, Finitely presented conjugacy-separable group|FULL LIST, MORE INFO

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