Substitution groups of formal power series

Abstract

Let $G$ be the group of power series $x+a_2{x}^2+a_3{x}^3+\cdots \in R[[x]]$ under substitution, where $R$ is a commutative ring with $1\ne 0$ of prime characteristic $p$. Given any $n\ge 1$, the subgroup $K_n=\{x+a_{n+1}{x}^{n+1}+a_{n+2}{x}^{n+2}+\cdots |a_i\in R\}$ is normal in $G$, and the quotient $G_n=G/K_n$ is the group of truncated polynomials over $R$ of degree $\le n$ under substitution. In this paper, we compute the exponent of the image of $K_r$ in $G_n$, for all $r,n\ge 1$, indicating in every case a family of elements realizing this exponent.