Burnside group

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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 $d \ge 3$
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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		$2^s3^t$ 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 $n \ge 2$ to avoid the free and trivial cases)	Value of $d$ (we assume $d \ge 1$ 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)	$2^{2728}$	?