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

TL;DR

This paper investigates the lower central series of the Riordan group over the field with two elements, revealing its structure and relations to other algebraic groups, and extends some results to more general rings.

Contribution

It computes the lower central series of the Riordan group over ${f F}_2$ and describes its abelianization over arbitrary rings with specific conditions.

Findings

Lower central series of ${f R}({f F}_2)$ is explicitly calculated.

The abelianization of the Riordan group over certain rings is characterized.

The structure relates to the Nottingham and Lagrange subgroups.

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.