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

Definition

The Burnside group $B(d,n)$ (sometimes called the free Burnside group) is defined as the quotient of the free group on $d$ generators by the normal subgroup generated by all $n^{th}$ powers. A Burnside group is a group that occurs as $B(d,n)$ for some choice of $d$ and $n$.

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, $B(d,n)$ is free in the subvariety of the variety of groups comprising those groups where $n^{th}$ 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 $n$ under which the Burnside groups $B(d,n)$ are all finite. For some small values of $n$, the Burnside groups are all finite, whereas for large enough values of $n$, the Burnside groups are all infinite for $d>1$.

Particular cases

Values of exponent

Value of $n$	What can we conclude about $B(d,n)$?	Order as a function of $d,n$	Nilpotency class in terms of $d,n$ (assume $d>0$)
0	finitely generated free group on $n$ generators	infinite	not nilpotent
1	trivial group, regardless of $d$	1	0
2	elementary abelian 2-group of rank $d$ and order $2^{d}$	$2^{d}$	1
3	2-Engel group with $n$ generators, exponent three	$3^{d+{\binom {d}{2}}+{\binom {d}{3}}}$	1 if $d=1$ 2 if $d=2$ 3 if $d\geq 3$
4	PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.	PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.	PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.
5	if finite, same as restricted Burnside group	if finite, then $5^{34}$ for $d=2$	12 for $d=2$ (if finite) 17 for $d=3$ (if finite)
6		$2^{s}3^{t}$ where $s=1+(d-1)3^{d+{\binom {d}{2}}+{\binom {d}{3}}},t=r+{\binom {r}{2}}+{\binom {r}{3}}$ where $r=1+(d-1)2^{d}$	not nilpotent

Value pairs

Value of $n$ (we assume $n\geq 2$ to avoid the free and trivial cases)	Value of $d$ (we assume $d\geq 1$ to avoid the trivial group case)	Group $B(d,n)$	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)	$3^{14}$	3
4	1	cyclic group:Z4	4	1
4	2	Burnside group:B(2,4)	4096	5
4	3	Burnside group:B(3,4)	$2^{69}$	?
4	4	Burnside group:B(4,4)	$2^{422}$	?
4	5	Burnside group:B(5,4)	$2^{2728}$	?

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 $B(d,0)$			|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