Nottingham group

In the mathematical field of infinite group theory, the Nottingham group is the group J(F_p) or N(F_p) consisting of formal power series t + a₂t²+... with coefficients in F_p. The group multiplication is given by formal composition also called substitution. That is, if

${\displaystyle f=t+\sum _{n=2}^{\infty }a_{n}t^{n}}$

and if ${\displaystyle g}$ is another element, then

${\displaystyle gf=f(g)=g+\sum _{n=2}^{\infty }a_{n}g^{n}}$.

The group multiplication is not abelian. The group was studied by number theorists as the group of wild automorphisms of the local field F_p((t)) and by group theorists including D. Johnson (1988) and the name "Nottingham group" refers to his former domicile.

This group is a finitely generated pro-p-group, of finite width. For every finite group of order a power of p there is a closed subgroup of the Nottingham group isomorphic to that finite group.