Burnside group

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This term is related to: combinatorial group theory
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Definition

The Burnside group $B(n,d)$ (sometimes called the free Burnside group) is defined as the quotient of the free group on $n$ generators by the normal subgroup generated by all $d^{th}$ powers. A Burnside group is a group that occurs as $B(n,d)$ 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, $B(n,d)$ is free in the subvariety of the variety of groups comprising those groups where $d^{th}$ powers are equal to the identity. In particular, any Burnside group is a group in which every fully invariant subgroup is verbal.

Particular cases

Value of $d$	What can we conclude about $B(n,d)$?	Order as a function of $n,d$	Nilpotency class in terms of $n,d$ (assume $n>0$)
0	finitely generated free group on $n$ generators	infinite	not nilpotent
1	trivial group, regardless of $n$	1	0
2	elementary abelian 2-group of rank $n$ and order $2^n$	$2^n$	1
3	2-Engel group with $n$ generators, exponent three	$3^{n+(\genfrac{}{}{0ex}{}{n}{2})+(\genfrac{}{}{0ex}{}{n}{3})}$	1 if $n=1$ 2 if $n=2$ 3 if $n\ge 3$
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5	unknown	unknown, but at least $5^6$	unknown, but at least 4 if finite

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 $B(n,0)$			|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