Lower central series of the Riordan group over the field with two elements

Abstract

The Riordan group ${\cal R}$ over the field ${\mathbb F}_2$ is a split extension of the Appell subgroup by the Nottingham group ${\cal N}({\mathbb F}_2)$. Using the lower central series of the Nottingham group obtained by C. Leedham-Green and S. McKay, the lower central series of ${\cal R}({\mathbb F}_2)$ is calculated. Considering the Riordan group over an arbitrary commutative ring with identity, where all Riordan arrays have only 1s on the main diagonal, it is also proved that the abelianization of this group is isomorphic to the direct product of the abelianization of the corresponding Lagrange subgroup and the additive group of the ground ring.