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 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", Australian Mathematical Society. Journal. Series A. Pure Mathematics and Statistics, 45 (3): 296–302, doi:10.1017/s1446788700031001, ISSN 0263-6115, MR 957195
Camina, Rachel (2000), "The Nottingham group", in du Sautoy, Marcus; Segal, Dan; Shalev, Aner, New horizons in pro-p groups, Progr. Math., 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", J. fuer die reine und angew. Math., 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, Progr. Math., 184, Boston, MA: Birkhäuser Boston, pp. 287–328, ISBN 978-0-8176-4171-9, MR 1765121

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Nottingham group

See also

References