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
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 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
Reduced free group |FULL LIST, MORE INFO