Restricted Burnside group

Definition

The restricted Burnside group $RB(d,n)$, sometimes denoted $R(d,n)$, is defined as the quotient group by the finite residual of the Burnside group $B(d,n)$. In other words, it is the quotient of $B(d,n)$ by the intersection of all normal subgroups of finite index in it.

Note that this group is a finite group if and only if the restricted Burnside problem for the pair $(d,n)$ has the answer Yes.

Facts

• Kostrikin's theorem on restricted Burnside problem: For any prime number $p$, the group $RB(d,p)$ is finite for every value of $d$.
• The condition that this group be finite is weaker than the condition that the Burnside group $B(d,n)$ be finite, i.e., there are many cases where $B(d,n)$ is known to be infinite and $RB(d,n)$ is known to be finite. This includes all odd primes greater than 665.
• In those cases where $B(d,n)$ is finite, $RB(d,n)$ is isomorphic to $B(d,n)$.