On the Lie algebra of a Burnside group of exponent $5$

Abstract:

A. Kostrikin proved that $R(5,2)$ has order at most 5$^{34}$ and is nilpotent of class at most 13. He verified this by showing that the freest Lie ring $L$ on 2 generators of characteristic 5 which satisfies the 4th Engel condition must be nilpotent of class at most 13. He then appealed to the correspondence between Lie rings and groups, given by the associated Lie ring of a group, to complete his proof. We studied Kostrikin’s calculations and considered a collection of matrices based on them. From these, we were able to construct $L$ with the use of a Univac 1107 computer and verify that $L$ is nilpotent of class exactly 13 with order exactly 5$^{34}$. Hence, modulo the conjecture made by Kostrikin, Sanov and others that $L$ is the associated Lie ring of $R(5,2)$, the order of $R(5,2)$ is exactly 5$^{34}$.