Finitely generated profinite 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 profinite group
View other group property conjunctions OR view all group properties
Contents
Definition
A finitely generated profinite group is a group that is both a finitely generated group and a profinite group.
Relation with other properties
Stronger properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
Finite group | finitely generated and profinite not implies finite | |FULL LIST, MORE INFO |
Weaker properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
Finitely generated residually finite group | finitely generated and residually finite | profinite implies residually finite | |FULL LIST, MORE INFO | |
Finitely generated Hopfian group | finitely generated and Hopfian | (via residually finite) | Finitely generated residually finite group|FULL LIST, MORE INFO | |
Residually finite group | Finitely generated residually finite group, Profinite group|FULL LIST, MORE INFO | |||
Hopfian group | Finitely generated residually finite group|FULL LIST, MORE INFO | |||
Finitely generated group | Finitely generated residually finite group|FULL LIST, MORE INFO | |||
Profinite group | |FULL LIST, MORE INFO |