Truncated Nottingham group

Name
This group may be referenced by any of the names truncated Nottingham group, truncated Johnson group, Nottingham polynomial group, Johnson polynomial group, or something similar.

Over a field
The truncated Nottingham group with parameter $$n$$ over a field $$K$$ is defined as the quotient of the defining ingredient::Nottingham group over $$K$$ (which, set-theoretically, is a subset of the formal power series ring $$Kt$$, by the relation of being congruent modulo $$t^{n+1}$$. Equivalently, it is the quotient of the Nottingham group by the normal subgroup comprising all those formal power series that are congruent to $$t$$ modulo $$t^{n+1}$$.

The group can be described as an algebraic group over $$K$$ and has dimension $$n - 1$$ as an algebraic group. The group also has a natural algebra group structure.

For a prime power
The truncated Nottingham group with parameter $$n$$ for a prime power $$q$$ can be defined as the truncated Nottingham group with parameter $$n$$ over the finite prime field $$\mathbb{F}_q$$. This is a group of order $$p^{n-1}$$.