Pro-p group ($P=\underleftarrow{\lim }{P}_i$)

A pro-$p$ group is an inverse limit of finite $p$-groups. Every continuous action on a finite set or finite module factors through a finite $p$-group. In particular its action on ${\mu }_p$ is trivial when the field characteristic differs from $p$, because $\mathrm{Aut}({\mu }_p)$ has order $p-1$.

A closed subgroup of a profinite group that is a pro-p group with the induced topology. Its images in finite continuous quotients are p-groups.

For a pro-p ring $R$, an $R$-analytic pro-p group is a pro-p group equipped with an analytic manifold structure locally modelled on open subsets of a finite free $R$-module, such that multiplication and inversion are given locally by convergent power series. The number of local coordinates is its analytic dimension. The Nottingham group admits no such structure over any pro-$p$ ring: its hereditary just infiniteness, unbounded finite cyclic torsion and sub-base-three derived-series growth exclude both alternatives in the analytic just-infinite structure theorem.

Over $k={\mathbb{F}}_p$, the Nottingham group is $\mathcal{N}(k)=\{f(t)=t+{\sum }_{i\ge 2}{a}_i{t}^i:a_i\in k\}$ 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.

Realize a finite p-group $P$ as the Galois group of a totally ramified extension $E/{\mathbb{F}}_p((t))$, using Artin-Schreier embedding problems over a characteristic-p field. A uniformizer $u$ of $E$ gives $E\cong {\mathbb{F}}_p((u))$: every element has a unique convergent Laurent expansion with coefficients in the residue field. An automorphism has $u\mapsto a_{1}u+a_2{u}^2+\cdots$, and $a_1$ defines a homomorphism $P\to {\mathbb{F}}_p^{\times }$. Its image is trivial because $|P|$ is a $p$-power and $p\nmid |{\mathbb{F}}_p^{\times }|$. Thus the faithful Galois action lies in the Nottingham group, embedding $P$.

Put ${\mathcal{N}}_m=\{f:f(t)\equiv t(\mathrm{mod}{t}^{m+1})\}$ for $m\ge 1$. These are open normal subgroups and ${\mathcal{N}}_m/{\mathcal{N}}_{m+1}\cong ({\mathbb{F}}_p,+)$ via the coefficient of $t^{m+1}$. Thus $[\mathcal{N}:{\mathcal{N}}_m]=p^{m-1}$. The depth of a nonidentity series is ${\mathrm{ord}}_t(f(t)-t)-1$. If $f=t+a{t}^{r+1}+\cdots$ and $g=t+b{t}^{s+1}+\cdots$, expansion of substitution givesHere $[f,g]=f^{-1}\circ g^{-1}\circ f\circ g$. This formula controls commutator subgroups and normal subgroups.

For odd $p$, group commutator collection in the depth filtration gives $[{\mathcal{N}}_m,{\mathcal{N}}_m]={\mathcal{N}}_{2m+1}$. Every degree $k\ge 2m+1$ occurs as a leading group commutator degree by choosing $r,s\ge m$, $r+s=k$, and $r\not\equiv s(\mathrm{mod}p)$; changing $r$ by one changes $r-s$ by two. Closed successive coefficient correction then fills the tail. Therefore ${\mathcal{N}}^{(n)}={\mathcal{N}}_{2^{n+1}-1}$ and ${\log }_p[\mathcal{N}:{\mathcal{N}}^{(n)}]=2^{n+1}-2$. In characteristic two the sharper collection theorem gives ${\log }_2[\mathcal{N}:{\mathcal{N}}^{(n)}]\le C{\kappa }^n$ for a constant $C$, with $\kappa =(1+\sqrt{17})/2<3$. In both cases the exponential base is strictly smaller than three.

If a pro-$p$ presentation on $d$ generators and possibly infinitely many relators $r_i$ satisfieswith the sum convergent and $D$ the Magnus degree of a free pro-p group element, then the presented pro-p group is infinite. The Golod-Shafarevich inequality forces its completed group algebra to be infinite-dimensional, which would be impossible for a finite group. This formulation applies to relators whose degrees tend to infinity and is suitable for enumerated power-relation constructions.

A pro-p group whose closed commutator subgroup lies in its closed power subgroup $G^{p^{ϵ}}$, with $ϵ=1$ for odd $p$ and $ϵ=2$ for $p=2$. This is the profinite analogue of a powerful p-group. Finitely generated torsion-free examples are exactly the uniform pro-p groups. The additive group of a finite product of p-adic integers is an abelian example.

The inverse limit of the quotients of an abstract group by its normal subgroups of finite $p$-power index. Maps to pro-p groups factor uniquely through the canonical dense image in this completion. Unlike profinite completion, only finite p-groups are used.

For finite $X$, this is the pro-p completion of the free group on $X$. Every function from $X$ to a pro-p group extends uniquely to a continuous group homomorphism. More generally one can require a basis tending to the identity and impose the universal property for maps with that convergence condition; alternatively the pro-$p$ completion of the abstract free group represents unrestricted maps from an arbitrary set. These infinite-rank conventions should not be conflated when interpreting generator cardinalities.

Embed a free pro-p group on $x_1,\dots ,x_d$ into the units of the noncommutative power-series algebra ${\mathbb{F}}_p⟨⟨X_1,\dots ,X_d⟩⟩$ by $x_i\mapsto 1+X_i$. Its Magnus degree is the lowest total degree occurring in $\iota (w)-1$, with $D(1)=\infty$. Since $(1+v{)}^{p^e}=1+v^{p^e}$ in characteristic $p$, one has $D(w^{p^e})\ge p^{e}D(w)$. High powers therefore impose relations with small weight at any fixed parameter $0<\tau <1$.

A finite generating set and finitely many relators in a free pro-p group, divided out by their closed normal closure. Relators are elements of the free pro-p group; they need not be finite ordinary words. A pro-p group with such a presentation is finitely presented in the pro-$p$ category.

Suppose $K$ is an open normal subgroup of a pro-p group $G$ and has a finite pro-p presentation. Choose lifts of generators of the finite quotient $G/K$. Add finitely many relators giving their conjugation and inverse conjugation on the generators of $K$, and replace each quotient relator by its value in $K$. The resulting finite finite pro-p presentation maps onto $G$. Its closed subgroup generated by the kernel generators is isomorphic to $K$, and its quotient by that subgroup is exactly $G/K$, so the map is also injective.

For a finitely generated pro-p group, its profinite Frattini subgroup has finite elementary abelian quotient. Continuous group homomorphisms to ${\mathbb{F}}_p$ are precisely the linear functionals on this quotient. The topological generation criterion identifies a minimal generating set with a basis, giving the displayed equality. Finite generation is essential when generator number means the least cardinality of an unrestricted dense generating family.

Let $I$ be the set of infinite binary sequences and $G={\mathbb{F}}_p^I$ with its product topology. Index the finite binary strings by the positive integers. For each $n$, take the element whose $r$ coordinate is one exactly when the $n$th string is a prefix of $r$. These countably many elements generate a dense subgroup: given finitely many distinct sequences, sufficiently long prefixes distinguish them individually. No finite family generates $G$, so its topological generator number is countable. Every continuous linear functional depends on finitely many coordinates, however, so $H^1(G,{\mathbb{F}}_p)$ has a basis of size $|I|=2^{{\aleph }_0}$. Thus the arbitrary-rank identity is false for unrestricted topological generating sets.

A finitely generated powerful p-group in the category of pro-p groups is uniform if all successive indices in its lower p-series are equal. Equivalently it is torsion-free. If its generator number is $d$, the successive indices are $p^d$, the power map carries $G^{p^n}$ bijectively onto $G^{p^{n+1}}$, and an ordered minimal generating set gives unique coordinates $g_1^{{\lambda }_1}\cdots g_d^{{\lambda }_d}$ with ${\lambda }_i$ in the p-adic integers. For $p=2$, powerfulness means $[G,G]\le G^4$.

For an odd prime $p$, the matrix logarithm and matrix exponential are inverse maps between $1+p{M}_d({\mathbb{Z}}_p)$ and $p{M}_d({\mathbb{Z}}_p)$. They imply that the $p$th-power map carries $U_m$ bijectively onto $U_{m+1}$. Direct multiplication gives $[U_i,U_j]\subseteq U_{i+j}$, so the group is a powerful pro-p group. The quotient $U_m/U_{m+1}$ is the additive matrix space over ${\mathbb{F}}_p$, with constant index $p^{d^2}$. The $d^2$ matrices $1+p{E}_{ij}$ topologically generate, by successive approximation through these quotients. The lower p-series is exactly $U_m$, so the group is a uniform pro-p group and a torsion-free group. At $p=2$ the first congruence subgroup contains $-I$ and is not uniform.

For odd $p$ and a finitely generated powerful pro-p group, put $U_j=G^{p^j}$. The displayed group homomorphisms are surjective. If one has a nonzero kernel, choose $x\in U_j\setminus U_{j+1}$ with $x^p\in U_{j+2}$. Successive corrections $x\mapsto x{u}_r$, with $u_r\in U_r$, cancel its $p$-th power one layer at a time, using $(x{u}_r{)}^p\equiv x^p{u}_r^p(\mathrm{mod}{U}_{r+2})$. Compactness and the separated power filtration give a nonidentity limit of order $p$. Thus a torsion-free group has constant layer indices and is a uniform pro-p group. Conversely constant indices make every displayed map injective; comparison at the first differing power layer proves that $x\mapsto x^p$ is injective, so there is no nontrivial torsion.

Every uniform pro-p group embeds continuously in ${\mathrm{GL}}_n({\mathbb{Z}}_p)$ for some finite $n$. Apply the Ado theorem to ${\mathbb{Q}}_p\otimes L_G$ and choose a deep power lattice whose representing matrices lie in $p\mathrm{End}(\Lambda )$, or $4\mathrm{End}(\Lambda )$ at $p=2$. The matrix exponential and logarithm give a faithful representation of the corresponding open normal power subgroup. Induce that representation to the whole group using its finitely many cosets. An element outside the open subgroup permutes the coset blocks nontrivially; an element inside it is detected by the original representation. Thus the induced representation is faithful. All block matrices preserve a finite free ${\mathbb{Z}}_p$-module, giving the integral embedding.

For a uniform pro-p group with $U_i=G^{p^i}$, the map $x\mapsto x^{p^a}$ transports congruences byCommutator collection gives $(yu{)}^p\equiv y^p{u}^p(\mathrm{mod}{U}_{b+2})$ for $u\in U_b$, and the power map is an isomorphism $U_b/U_{b+1}\to U_{b+1}/U_{b+2}$. Thus a nontrivial first difference moves down exactly one layer when taking a $p$th power. Iterating gives the equivalence; completeness supplies unique roots when the congruence is read in reverse.

For a uniform pro-p group, the power map $x\mapsto x^{p^n}$ is a homeomorphism onto $G^{p^n}$. Transport multiplication back to $G$ byUnique roots make this an associative group operation. For odd $p$, group commutator collection gives $x{+}_{n}y\equiv y{+}_{n}x(\mathrm{mod}{G}^{p^{n+1}})$ and $x{+}_{n}y\equiv x{+}_{n-1}y(\mathrm{mod}{G}^{p^n})$. The operations converge to addition in the Lie algebra of a uniform pro-p group.

The Lie algebra of a uniform pro-p group construction and BCH convergence on a powerful Lie lattice are mutually inverse on objects and morphisms. On the group side morphisms are continuous group homomorphisms; on the lattice side they are linear Lie algebra homomorphisms over the p-adic integers. A group homomorphism respects powers, roots and limits, and a Lie homomorphism respects every convergent Baker--Campbell--Hausdorff formula term.

The Lie algebra of a uniform pro-p group construction and BCH convergence on a powerful Lie lattice are mutually inverse on objects and morphisms. On the group side morphisms are continuous group homomorphisms; on the lattice side they are linear Lie algebra homomorphisms over the p-adic integers. A group homomorphism respects powers, roots and limits, and a Lie homomorphism respects every convergent Baker--Campbell--Hausdorff formula term.

Every abstract subgroup of finite index in a finitely generated pro-p group is open. Fixed-generator commutator collection in nilpotent groups makes the abstract commutator subgroup closed. The resulting abelian quotient is a quotient of a finite power of the p-adic integers, so its subgroup of $p$th powers is closed and of finite index. Thus the algebraic $G^p[G,G]$ is open and every abstract index-$p$ subgroup is open. An arbitrary finite quotient is a finite p-group; induction on its order, using a normal subgroup of index $p$ and the Schreier lemma, gives the assertion. The normal core reduces nonnormal finite-index subgroups to this case.

A Sylow pro-$p$ subgroup of a profinite group is a closed pro-$p$ subgroup whose images are Sylow $p$-subgroups in every finite quotient. Such subgroups exist and are conjugate. Every open subgroup containing one has index prime to $p$. They allow prime-to-$p$ finite extensions to detect $p$-primary Galois cohomology.