Graduate Seminar

Location:  CL 312 and Live Stream

Speaker: Agustin D'Alessandro

MSc Student supervised by Fernando Szecthman

Title:  The group of formal power series under substitution

 Zoom Link:  https://uregina-ca.zoom.us/j/92805435386?pwd=42pEMT2QL4b8W8bqIOr5Y39hln8bSI.1

Abstract:

Given a commutative ring with identity \(R\), the subset \(G(R)\) of \(R[[x]]\) of the form
\[
\alpha = x+a_2x^2+a_3x^3+\dots
\]
is a group under substitution. It was first studied by S. Jennings, D. L. Johnson and I. O. York. After their investigation, and as the interest in the theory of pro-\(p\) groups continued to grow, this group began to attract more attention, especially when \(R\) is a finite field of characteristic \(p\), a prime number. In such cases, \(G(R)\) is also known as the Nottingham group, denoted \(\mathcal{N}(R)\).

In this lecture, we give a construction of \(G(R)\) and go through many basic results from Jennings, Johnson and York, which we seek to extend. We give a matrix representation of \(G(R)\), first described by York, which is very useful when computing inside this group. The very construction of the group gives rise to many normal subgroups which produce interesting quotient groups. When these quotient groups are finite, we are interested in obtaining a power-commutator presentation. For that purpose, we explore both the commutator and the power structure of this group.