Finitely presented periodic group
From Groupprops
This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: finitely presented group and periodic group
View other group property conjunctions OR view all group properties
Contents
Definition
A group is termed a finitely presented periodic group if it satisfies both the following conditions:
- It is a finitely presented group -- it has a presentation that uses a finite number of generators and a finite number of relations.
- It is a periodic group -- every element has finite order.
Conjecture
The conjecture that every finitely presented periodic group is finite is currently open. Note that there do exist finitely generated periodic groups that are not finite, such as the Grigorchuk group and Tarski monsters and other negative solutions to the Burnside problem (see periodic not implies locally finite).
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| finite group | underlying set is finite | obvious | implication conjectured to not be strict, i.e., the properties are conjectured to be equal | -- |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| finitely presented group | has a finite presentation | |FULL LIST, MORE INFO | ||
| periodic group | every element has finite order | Finitely generated periodic group|FULL LIST, MORE INFO | ||
| finitely generated periodic group | finitely generated and periodic | |FULL LIST, MORE INFO |
