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

2025/11/26 by Nikolai A. Krylov, Krylov, Nikolai A.
Mathematics · #20D15 #20E18 #Advanced Combinatorial Mathematics #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.2511.21639

openalex publication_date 2025/11/26 · openalex created_date 2025/11/28 · openalex updated_date 2026/07/28

Abstract

The Riordan group \cal R over the field \mathbb F₂ is a split extension of the Appell subgroup by the Nottingham group \cal N(\mathbb F₂). 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 F₂) 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.

Citations

• On embeddability of Coxeter groups into the Riordan group
• Commutators and commutator subgroups of the Riordan group
• Some Open Questions about Random Walks, Involutions, Limiting Distributions, and Generating Functions

Related

• Centralizers of the Riordan Group
• Finding an Isomorphism between the Riordan Group and a Subgroup of the Double Riordan Group
• Riordan Groups in higher Dimensions
• Roots of Formal Power Series and New Theorems on Riordan Group Elements
• On the Central Description of the Group of Riordan Arrays