Profinite group

A profinite group is an inverse limit of finite discrete groups. Equivalently, it is a compact Hausdorff totally disconnected topological group.

An infinite profinite group is just infinite if every nontrivial closed normal subgroup has finite index, equivalently is open. Requiring closedness is part of the topological convention. This property says that every proper Hausdorff quotient is finite.

An infinite profinite group is hereditarily just infinite if every open subgroup is a just-infinite profinite group. To prove this, it suffices to show that every nonidentity element of every open subgroup has open closed normal closure: any nontrivial closed normal subgroup then contains that normal closure. The Nottingham group is an example. In odd characteristic its depth-commutator formula constructs nonzero elements of each sufficiently high layer inside any such normal subgroup, and successive coefficient correction fills the entire tail.

The rank of a profinite group is $\mathrm{rk}(G)={\sup }_{K{\le }_{c}G}d(K)$, where $K$ runs through closed subgroups and $d(K)$ is the least number of topological generators. If an open subgroup $H$ has rank $r$, its normal core $C$ is open and has rank at most $r$. Every closed $K\le G$ has a normal subgroup $K\cap C$ generated by at most $r$ elements, while $K/(K\cap C)$ has order at most $[G:C]$. A finite group of order at most $M$ is generated by at most $\lfloor {\log }_{2}M\rfloor$ elements, since each necessary new generator at least doubles the generated subgroup. Lifting generators yields $d(K)\le r+\lfloor {\log }_2[G:C]\rfloor$. Thus finite rank is preserved in both directions under passage to an open subgroup.

A profinite group $G$ is procyclic if every quotient by an open normal subgroup is cyclic. Equivalently $G=\overline{⟨g⟩}$ for one element $g$. For the forward implication, the sets of lifts of generators of $G/N$ are nonempty clopen sets. Any finite intersection contains a lift of a generator of the cyclic quotient by the intersection of those open normal subgroups. Compactness supplies one element generating every finite quotient, and hence generating densely. The reverse implication follows by projecting that generator. For a pro-p group, two distinct maximal open subgroups would give a quotient $C_p^2$, so a nontrivial procyclic group has a unique maximal open subgroup $M$. Its profinite Frattini subgroup is $M$, and any $g\notin M$ generates topologically: otherwise the proper closed subgroup it generates lies in a maximal open subgroup. The additive group of p-adic integers is an infinite example, generated topologically by $1$.

Every identity neighbourhood of a compact Hausdorff totally disconnected topological group contains an open normal subgroup. Choose a clopen set $U$ containing the identity inside that neighbourhood. Compactness and continuity give a symmetric identity neighbourhood $V$ with $VU\subseteq U$, hence $VU=U$. The setwise stabilizer $H=\{g:gU=U\}$ is an open subgroup contained in $U$. Its index is finite, so its normal core is an open normal subgroup contained in $U$. These subgroups recover the profinite group as the inverse limit of its finite quotients.

The intersection of the maximal proper open subgroups of a profinite group. A subset topologically generates the profinite group exactly when its image topologically generates its quotient by this intersection: a proper closed generated subgroup is contained in a maximal open subgroup, obtained from a suitable finite quotient. For a pro-p group, $\Phi (G)=\overline{G^p[G,G]}$, where the bar denotes closure of the algebraically generated power and commutator subgroups. Closedness cannot simply be discarded in infinite rank.

A profinite abelian group is an inverse limit of finite abelian groups. Its topology is compact and totally disconnected. An abelian pro-p group acquires scalar multiplication by p-adic integers by taking limits of integer multiples.

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.

The profinite topology on a group takes the cosets of finite-index normal subgroups as a neighborhood basis. A profinite group carries the inverse-limit topology from its finite quotients.

The profinite topology on $\mathbb{Z}$ has basis the residue classes $a+k\mathbb{Z}$, $k\ge 1$. These classes are both open and closed, since their complements are finite unions of other residue classes. Set $F_n=(4^n-1)/3+4^n\mathbb{Z}$ for $n\ge 1$. Then $F_{n+1}\subseteq F_n$ and every $F_n$ is nonempty. An integer $x$ in their intersection would make $3x+1$ divisible by every $4^n$, forcing $3x+1=0$, impossible. Thus the open complements $\mathbb{Z}\setminus F_n$ form an increasing cover with no finite subcover. The topology is Hausdorff because distinct integers are distinct modulo a sufficiently large integer, but is not compact.

An inverse limit carries the coarsest topology making every coordinate projection map continuous. It is the subspace topology inherited from the product of the component spaces.

The profinite completion of a group $G$ is the inverse limit of its finite quotient groups:The canonical map $G\to \hat{G}$ has dense image and is injective exactly when $G$ is residually finite.

Every homomorphism from a group $G$ to a profinite group $P$ extends uniquely to a continuous homomorphism $\hat{G}\to P$. Construct the extension on every finite quotient of $P$ and invoke the universal property of an inverse limit; uniqueness follows from the density of $G$ in $\hat{G}$.

A subset $S$ of a topological group topologically generates it when the ordinary subgroup $⟨S⟩$ is dense.

An element of a topological generating set of a topological group. A finite tuple consists of topological generators when the abstract subgroup it generates is dense in the group; it need not equal that abstract subgroup.

A topological group is topologically finitely generated when it has a finite topological generating set.

For a profinite group $G={\underleftarrow{\lim }}_j{G}_j$, a subset $S\subseteq G$ is a topological generating set exactly when $⟨p_j(S)⟩=p_j(G)$ in every finite quotient. Indeed, a subgroup of a profinite group is dense exactly when its image in every finite continuous quotient is surjective.

Elements $g,h$ of a profinite group $G={\underleftarrow{\lim }}_j{G}_j$ are conjugate exactly when their images are conjugate in every $G_j$. For the nontrivial direction, the finite nonempty sets of conjugators in the $G_j$ form an inverse system, whose inverse limit supplies a conjugator in $G$.

A topological group has the topological Hopf property when every continuous surjective endomorphism is a topological automorphism.

Every topologically finitely generated group that is profinite has the topological Hopf property. There are only finitely many open subgroups of each given index, and inverse image under a surjective endomorphism permutes them. The endomorphism kernel consequently lies in every open normal subgroup and is trivial.

For $G={\underleftarrow{\lim }}_j{G}_j$, the equation $x^m=g$ has a solution in $G$ exactly when it has a solution after projection to every $G_j$. The finite nonempty sets of roots form an inverse system, and a compatible family is a root in $G$.