Commutators and commutator subgroups of the Riordan group

TL;DR

This paper investigates the structure of the Riordan group by calculating its derived series and commutator subgroups, providing new insights into its subgroup hierarchy and related formal power series groups.

Contribution

It introduces a method to compute the n-th commutator subgroup of the Riordan group's associated subgroup, advancing understanding of its algebraic structure.

Findings

Derived series of the Riordan group calculated

n-th commutator subgroup of the associated subgroup determined

Results applied to groups of formal power series

Abstract

We calculate the derived series of the Riordan group. To do that, we study a nested sequence of its subgroups, herein denoted by $\mathcal G_k$. By means of this sequence, we first obtain the n-th commutator subgroup of the Associated subgroup. This fact allows us to get some related results about certain groups of formal power series and to complete the proof of our main goal, Theorem 1 in this paper.