On the abelianization of certain groups of formal power series

TL;DR

This paper computes the abelianization of a specific group of formal power series with certain coefficient restrictions, extending previous work by providing a direct approach for general k.

Contribution

It offers a new direct method to compute the abelianization of Jennings groups of formal power series for arbitrary k, beyond the previously known case k=2.

Findings

Explicit abelianization of Jennings groups for general k

New direct computational approach for abelianizations

Extension of previous results to broader class of power series groups

Abstract

We compute the abelianization of the Jennings group $\mathcal{J}_k(\mathbb{Z})$ of powers series with constant coefficient $0$, linear coefficent equal to $1$ and vanishing coefficients in orders greater or equal than $2$ and less than $k$, where $k\geqslant2$. This is accomplished by directly dealing with the equivalence classes in the corresponding abelianizations, in contrast with the work of I. K. Babenko and S. A. Bogatyy, who give an explicit abelianization morphism for the case $k=2$.