Nottingham group

Definition
This group is variously called the Johnson group, Nottingham group, or group of formal power series under substitution. For truncated versions of this group, see truncated Nottingham group.

Definition for a field
Let $$K$$ be a field. As a set, the Nottingham group over $$K$$ is the set of all formal power series with no constant term and with the coefficient of the degree 1 term equal to 1. If we denote the formal variable by $$t$$, it is the set of formal power series of the form:

$$t + a_2t + a_3t^2 + \dots + a_nt^n + \dots, \mbox{ all } a_i \in K $$

The group operation is by composition. In other words, for power series $$f,g$$, the product is defined as $$f \circ g$$: the formal power series obtained by plugging in $$g$$ for $$t$$ in the expression for $$f$$.

The two-sided identity element is the power series:

$$\! t$$

It can be checked that the operation is associative (basically, this is related to the fact that function composition is associative) and that every element has an inverse with respect to the operation.

Definition for a prime number
Let $$p$$ be a prime number. The Nottingham group corresponding to $$p$$ is defined as the Nottingham group over the finite prime field $$\mathbb{F}_p$$.