Burnside group
This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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This term is related to: combinatorial group theory
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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 ![]() ![]() |
---|---|---|---|
0 | finitely generated free group on ![]() |
infinite | not nilpotent |
1 | trivial group, regardless of ![]() |
1 | 0 |
2 | elementary abelian 2-group of rank ![]() ![]() |
![]() |
1 |
3 | 2-Engel group with ![]() |
![]() |
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 ![]() ![]() |
12 for ![]() 17 for ![]() |
6 | ![]() ![]() ![]() |
not nilpotent |
Value pairs
Value of ![]() ![]() |
Value of ![]() ![]() |
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 |