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

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

and if $g$ is another element, then

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

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.

References

• Johnson, D. L. (1988), "The group of formal power series under substitution", Journal of the Australian Mathematical Society, Series A, 45 (3): 296–302, doi:10.1017/s1446788700031001, ISSN 0263-6115, MR 0957195
• Camina, Rachel (2000), "The Nottingham group", in du Sautoy, Marcus; Segal, Dan; Shalev, Aner (eds.), New horizons in pro-p groups, Progress in Mathematics, vol. 184, Boston, MA: Birkhäuser Boston, pp. 205–221, ISBN 978-0-8176-4171-9, MR 1765121
• Fesenko, Ivan (1999), "On just infinite pro-p-groups and arithmetically profinite extensions", Journal für die reine und angewandte Mathematik, 517: 61–80
• du Sautoy, Marcus; Fesenko, Ivan (2000), "Where the wild things are: ramification groups and the Nottingham group", in du Sautoy, Marcus; Segal, Dan; Shalev, Aner (eds.), New horizons in pro-p groups, Progress in Mathematics, vol. 184, Boston, MA: Birkhäuser Boston, pp. 287–328, ISBN 978-0-8176-4171-9, MR 1765121