Lie algebra associated with the group of finitary automorphisms of p-adic tree

We study the Sylow p-subgroup FP∞ of the finitary automorphism group of the p-adic rooted tree for a prime p>2. The fully invariant subgroups of the group FP∞ are characterized, in particular the lower central series and the Lazard series of FP∞ are described. This allows us to study the Lie algebra associated with the group FP∞ and show that it has a special tableau representation. The lower central series, the derived series, and the Engel series of this Lie algebra are described.