Finitely generated periodic 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 periodic group
View other group property conjunctions OR view all group properties
Contents
Definition
Symbol-free definition
A finitely generated periodic group is a group satisfying the following two conditions:
- It is finitely generated -- it has a generating set of finite cardinality.
- It is periodic -- every element in the group has finite order.
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| Finite group | Follows from periodic not implies locally finite (see also examples: Grigorchuk group, infinite Burnside groups, and Tarski monsters) | Finitely presented periodic group|FULL LIST, MORE INFO | ||
| Finitely presented periodic group | both periodic and finitely presented; conjectured to be equivalent to being a finite group | |FULL LIST, MORE INFO |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| Periodic group | |FULL LIST, MORE INFO | |||
| Finitely generated group | |FULL LIST, MORE INFO | |||
| Group generated by finitely many periodic elements | |FULL LIST, MORE INFO |
