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 |
