Elements of Finite Order in the Group of Formal Power Series Under Composition

TL;DR

This paper studies elements of finite order in the group of formal power series under composition, proving conjugacy to linear terms and constructing elements of specific orders with unique coefficient sequences.

Contribution

It provides a characterization of finite order elements and a method to construct such elements with prescribed properties, including possibly new proofs.

Findings

Finite order elements are conjugate to their linear part.

Construction of order n elements given a primitive n-th root of unity.

Unique sequences of coefficients determine elements of a given finite order.

Abstract

We consider formal power series $f(z) = \omega z + a_2z^2 + \ldots \ (\omega \neq 0)$, with coefficients in a field of characteristic $0$. These form a group under the operation of composition (= substitution). We prove (Theorem 1) that every element $f(z)$ of finite order is conjugate to its linear term $\ell_\omega(z) = \omega z$, and we characterize those elements which conjugate $f(z)$ to $\omega z$. Then we investigate the construction of elements of order $n$ and prove (Theorem 2) that, given a primitive $n$'th root of unity $\omega$ and an arbitrary sequence $\{a_k\}_{k\neq nj+1}$ there is a unique sequence $\{a_{nj + 1}\}_{j=1}^\infty$ such that the series $f(z) = \omega z + a_2z^2 + a_3z^3 + \ldots$ has order $n$. Sections 1 - 5 give an exposition of this classical subject, written for the 2005 - 2006 Morgan State University Combinatorics Seminar. We do not claim priority for these results in this classical field, though perhaps the proof of Theorem 2 is new. We have now (2018) added Section 6 which gives references to valuable articles in the literature and historical comments which, however incomplete, we hope will give proper credit to those who have preceded this note and be helpful and of interest to the reader.