• Source: Nottingham group

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




f
=
t
+

∑

n
=
2


∞



a

n



t

n




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


and if



g


{\displaystyle g}

is another element, then




g
f
=
f
(
g
)
=
g
+

∑

n
=
2


∞



a

n



g

n




{\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 Fp((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

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