Burnside group

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.