Normal Subgroups in Substitution Groups of Formal Power Series

Abstract

Let R be a commutative ring with 1, and let N(R)≔t+t2R[t] be the group of normalized formal power series over R under substitution. In this paper we investigate the connection between the ideal structure of R and the normal subgroup structure of N(R). In particular, we show that, if K is a finite field of characteristic not equal to two, then every proper quotient group of the so-called Nottingham group N(K) is finite. As a further application we consider the profinite completion of the group N(R). We show that, if every additive subgroup of finite index in R contains an ideal of finite index in R, then N(R̂)≅N(R̂).