Nottingham group ( $N (F_{p})$ )

Over

k = F_{p}

, the Nottingham group is

N (k) = {f (t) = t + \sum_{i \geq 2} a_{i} t^{i} : a_{i} \in k}

with composition and the

t

-adic topology. Equivalently it is the group of continuous

k

-automorphisms of the Laurent series field

k ((t))

that act trivially on

(t) / (t^{2})

. With ordinary series composition, the isomorphism sends

f

to the automorphism

h (t) \mapsto h (f^{- 1} (t))

; this inverse avoids reversing the group product. It is a pro-p group, recovered from its finite truncations, and contains elements of arbitrarily large finite

p

-power order.

Table of contents
- Finite p-group embedding in the Nottingham group Nottingham group
- Depth filtration of the Nottingham group Nottingham group
  - Derived-series growth of the Nottingham group Depth filtration of the Nottingham group

Finite p-group embedding in the Nottingham group

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Realize a finite p-group

P

as the Galois group of a totally ramified extension

E / F_{p} ((t))

, using Artin-Schreier embedding problems over a characteristic-p field. A uniformizer

u

E

gives

E ≅ F_{p} ((u))

: every element has a unique convergent Laurent expansion with coefficients in the residue field. An automorphism has

u \mapsto a_{1} u + a_{2} u^{2} + \dots

, and

a_{1}

defines a homomorphism

P \to F_{p}^{\times}

. Its image is trivial because

∣ P ∣

is a

p

-power and

p ∤ ∣ F_{p}^{\times} ∣

. Thus the faithful Galois action lies in the Nottingham group, embedding

P

Depth filtration of the Nottingham group ( $N_{m} = {f : f (t) \equiv t (mod t^{m + 1})}$ )

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Put

N_{m} = {f : f (t) \equiv t (mod t^{m + 1})}

for

m \geq 1

. These are open normal subgroups and

N_{m} / N_{m + 1} ≅ (F_{p}, +)

via the coefficient of

t^{m + 1}

. Thus

[N : N_{m}] = p^{m - 1}

. The depth of a nonidentity series is

ord_{t} (f (t) - t) - 1

. If

f = t + a t^{r + 1} + \dots

and

g = t + b t^{s + 1} + \dots

, expansion of substitution gives

[f, g] (t) = t + ab (r - s) t^{r + s + 1} + O (t^{r + s + 2}) .

(1)

Here

[f, g] = f^{- 1} \circ g^{- 1} \circ f \circ g

. This formula controls commutator subgroups and normal subgroups.

Derived-series growth of the Nottingham group ( $lo g_{p} [N : N^{(n)}] = O (λ^{n}), λ < 3$ )

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For odd

p

, group commutator collection in the depth filtration gives

[N_{m}, N_{m}] = N_{2 m + 1}

. Every degree

k \geq 2 m + 1

occurs as a leading group commutator degree by choosing

r, s \geq m

r + s = k

, and

r \neq \equiv s (mod p)

; changing

r

by one changes

r - s

by two. Closed successive coefficient correction then fills the tail. Therefore

N^{(n)} = N_{2^{n + 1} - 1}

and

lo g_{p} [N : N^{(n)}] = 2^{n + 1} - 2

. In characteristic two the sharper collection theorem gives

lo g_{2} [N : N^{(n)}] \leq C κ^{n}

for a constant

C

, with

κ = (1 + 17) /2 < 3

. In both cases the exponential base is strictly smaller than three.

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The Nottingham Group refers to a research consortium or collective of researchers based at the University of Nottingham, primarily focusing on various fields such as health, education, and social sciences. The group often collaborates on projects related to public health, clinical research, and innovations in education, among other areas. One of the notable subsets associated with the Nottingham Group is the **Nottingham University Hospitals NHS Trust**, which collaborates with the university for clinical research and advancements in healthcare.

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Nottingham group ( $N (F_{p})$ )

Finite p-group embedding in the Nottingham group

Depth filtration of the Nottingham group ( $N_{m} = {f : f (t) \equiv t (mod t^{m + 1})}$ )

Derived-series growth of the Nottingham group ( $lo g_{p} [N : N^{(n)}] = O (λ^{n}), λ < 3$ )

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Nottingham group (N(Fp​))

Finite p-group embedding in the Nottingham group

Depth filtration of the Nottingham group (Nm​={f:f(t)≡t(modtm+1)})

Derived-series growth of the Nottingham group (logp​[N:N(n)]=O(λn),λ<3)

 Ancestors (8)

 Incoming links (6)

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 Articles by others on the same topic (1)

Nottingham group ( $N (F_{p})$ )

Depth filtration of the Nottingham group ( $N_{m} = {f : f (t) \equiv t (mod t^{m + 1})}$ )

Derived-series growth of the Nottingham group ( $lo g_{p} [N : N^{(n)}] = O (λ^{n}), λ < 3$ )

 Discussion (0)