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
- [math]\displaystyle{ f = t+ \sum_{n=2}^\infty a_n t^n }[/math]
and if [math]\displaystyle{ g }[/math] is another element, then
- [math]\displaystyle{ gf = f(g) = g+ \sum_{n=2}^\infty a_n g^n }[/math].
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, http://anziamj.austms.org.au/JAMSA/V45/Part3/Johnson.html
- Camina, Rachel (2000), "The Nottingham group", in du Sautoy, Marcus; Segal, Dan; Shalev, Aner, New horizons in pro-p groups, Progress in Mathematics, 184, Boston, MA: Birkhäuser Boston, pp. 205–221, ISBN 978-0-8176-4171-9, https://books.google.com/books?isbn=0817641718
- 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, New horizons in pro-p groups, Progress in Mathematics, 184, Boston, MA: Birkhäuser Boston, pp. 287–328, ISBN 978-0-8176-4171-9, https://books.google.com/books?isbn=0817641718
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