Lower central series of the Riordan group over the field with two elements
Nikolai A. Krylov
TL;DR
This work determines the lower central series of the Riordan group over the field ${\mathbb F}_2$ by leveraging the established lower central series of the Nottingham group and an inverse-limit perspective on Riordan groups. It proves that ${\gamma_n({\cal R}) \cong {\cal A}_{2n-3} \ltimes {\gamma_n({\cal N})}}$, with explicit lower central quotients given by ${\gamma_i({\cal R})/\gamma_{i+1}({\cal R}) \cong (\mathbb{Z}_2)^3 \times \mathbb{Z}_4}$ for $i=1$, $(\mathbb{Z}_2)^4$ for even $i>1$, and $(\mathbb{Z}_2)^6$ for odd $i>1$. It also describes the abelianization of the S-version of the Riordan group in general, and specializes to ${S{\cal R}^{ab}}(\mathbb{Z}) \cong \mathbb{Z}^3 \times (\mathbb{Z}_2)^2$, answering related questions about commutator subgroups. Additionally, the paper provides explicit embeddings of dihedral groups $D_{2^{n+1}}$ into truncated Riordan groups $T{\cal R}_{2^n}$, illustrating concrete finite-2-group realizations within the Riordan framework.
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.