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Automorphisms of the Nottingham group. (English) Zbl 0965.20021

Let \(K\) be a finite field of characteristic \(p\). The Nottingham group over \(K\) can be defined as the group \({\mathcal N}(K):=t+t^2K[[t]]\) of normalised power series under substitution. It is a finitely-generated pro-\(p\) group with some interesting properties [see the survey article in the book “New horizons in pro-\(p\) groups”, Birkhäuser, Prog. Math. 184, 205-221 (2000; Zbl 0977.20020)].
In the paper under review the author begins by giving representatives of the conjugacy classes of elements of order \(p\) in \({\mathcal N}(K)\). He then goes on to prove that, for \(p\geq 5\), every automorphism of \({\mathcal N}(K)\) is the composition of an inner automorphism and a field automorphism.
The Nottingham group can also be regarded as a group of automorphisms of the field \(K((t))\). Alternative proofs, from this viewpoint, are given in the Appendix.

MSC:

20F28 Automorphism groups of groups
20E18 Limits, profinite groups
20E36 Automorphisms of infinite groups
20E45 Conjugacy classes for groups

Citations:

Zbl 0977.20020
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References:

[1] R. Camina, Subgroups of the Nottingham group, Ph.D. thesis, London, 1996.; R. Camina, Subgroups of the Nottingham group, Ph.D. thesis, London, 1996. · Zbl 0883.20015
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