Burnside group
This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions
This term is related to: combinatorial group theory
View other terms related to combinatorial group theory | View facts related to combinatorial group theory
Contents
Definition
The Burnside group (sometimes called the free Burnside group) is defined as the quotient of the free group on generators by the normal subgroup generated by all powers. A Burnside group is a group that occurs as for some choice of and .
Note that any Burnside group is a reduced free group because it is a quotient group of a free group by a verbal subgroup. More explicitly, is free in the subvariety of the variety of groups comprising those groups where powers are equal to the identity. In particular, any Burnside group is a group in which every fully invariant subgroup is verbal.
Relation with Burnside problem
Further information: Burnside problem
The Burnside problem is the problem of determining the conditions on under which the Burnside groups are all finite. For some small values of , the Burnside groups are all finite, whereas for large enough values of , the Burnside groups are all infinite for .
Particular cases
Values of exponent
Value of | What can we conclude about ? | Order as a function of | Nilpotency class in terms of (assume ) |
---|---|---|---|
0 | finitely generated free group on generators | infinite | not nilpotent |
1 | trivial group, regardless of | 1 | 0 |
2 | elementary abelian 2-group of rank and order | 1 | |
3 | 2-Engel group with generators, exponent three | 1 if 2 if 3 if | |
4 | PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] | PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] | PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] |
5 | if finite, same as restricted Burnside group | if finite, then for | 12 for (if finite) 17 for (if finite) |
6 | where where | not nilpotent |
Value pairs
Value of (we assume to avoid the free and trivial cases) | Value of (we assume to avoid the trivial group case) | Group | Order | Nilpotency class |
---|---|---|---|---|
2 | 1 | cyclic group:Z2 | 2 | 1 |
2 | 2 | Klein four-group | 4 | 1 |
2 | 3 | elementary abelian group:E8 | 8 | 1 |
2 | 4 | elementary abelian group:E16 | 16 | 1 |
3 | 1 | cyclic group:Z3 | 3 | 1 |
3 | 2 | unitriangular matrix group:UT(3,3) | 27 | 2 |
3 | 3 | Burnside group:B(3,3) | 2187 | 3 |
3 | 4 | Burnside group:B(4,3) | 3 | |
4 | 1 | cyclic group:Z4 | 4 | 1 |
4 | 2 | Burnside group:B(2,4) | 4096 | 5 |
4 | 3 | Burnside group:B(3,4) | ? | |
4 | 4 | Burnside group:B(4,4) | ? | |
4 | 5 | Burnside group:B(5,4) | ? |
Relation with other properties
Stronger properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
Finitely generated free group | Burnside group | |FULL LIST, MORE INFO |
Weaker properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
Finitely generated group | ||||
Reduced free group | |FULL LIST, MORE INFO |