On the abelianization of certain groups of formal power series

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 $k⩾2$. 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 $k=2$.