A power-commutator presentation for the group of truncated polynomials under substitution

Abstract

Given a prime $p$, the Nottingham group $G(p)$ is the group of formal power series $x+a_2x^2+a_3x^3+\cdots$, $a_i\in{\bf Z}/p{\bf Z}$, under substitution. For $n\geq 1$, we write $G_n(p)$ for the quotient of $G(p)$ by its normal subgroup $K_n=\{x+a_{n+1}x^{n+1}+a_{n+2}x^{n+2}+\cdots\,|\, a_i\in {\bf Z}/p{\bf Z}\}$, namely the group of truncated polynomials $x+a_2x^2+\cdots+a_nx^n$, $a_i\in {\bf Z}/p{\bf Z}$, under substitution, which is a $p$-group of order $p^{n-1}$ generated by $x+x^2,\dots,x+x^n$. In this paper, we determine the power-commutator presentation of $G_{15}(p)$ relative to $x+x^2,\dots,x+x^{15}$. This, in turn, automatically gives the power-commutator presentation of $G_{n}(p)$ relative to $x+x^2,\dots,x+x^{n}$ for any $1\leq n<15$, as well as the initial segments of the power and commutator relations in $G_{n}(p)$ corresponding to $x+x^2,\dots,x+x^{15}$ for any $n>15$.