# 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 $d \ge 3$
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) $2^{2728}$  ?

## 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