Profinite Groups and the Krull Topology

Introduction

A profinite group is a topological group that is the inverse limit of an inverse system of finite groups, each finite group carrying the discrete topology. Equivalently, and this is the form in which the theory is usually applied, a profinite group is a compact Hausdorff totally disconnected topological group. The class is closed under the operations that matter — closed subgroups, quotients by closed normal subgroups, products and inverse limits — and it is the home of the Galois groups of infinite algebraic extensions, of the ring $\mathbb{Z}_p$ of $p$-adic integers and of its units, of the pro-$p$ groups of combinatorial group theory, and of the profinite completions of abstract groups.

The topology of a profinite group is not a convenience: it is the structure that makes the Galois correspondence a theorem about closed subgroups rather than all subgroups, and it is the reason the Galois group of an infinite extension need not be metrisable. The Krull topology, introduced for the Galois group of an arbitrary Galois extension, is the special case of the inverse-limit topology in which the finite groups are the Galois groups of the finite Galois subextensions. That topology is metrisable exactly when there are countably many such subextensions, so it is not induced by any distance for a field with uncountably many finite Galois subextensions; the article exhibits such a group explicitly.

The article develops the inverse limit and its topology, the dictionary of open and closed subgroups, the Krull topology and the Galois correspondence, the profinite completion of an abstract group, and the beginnings of the structure theory — pro-$p$ subgroups, the Frattini subgroup and generation. The abelian theory of the previous articles is used where it is convenient: a profinite abelian group is the Pontryagin dual of a discrete torsion group, a fact from Pontryagin Duality, and its structure is read off from the classification of discrete torsion groups. The Galois theory that the Krull topology topologises is that of Galois Theory and Splitting Fields and Algebraic Closure in Part I; the Haar measure of a compact group is standard, and the invariant integration of a profinite group, including its $L^p$ theory, belongs to Part III, where the measure and the limit are available. No physics is invoked.

Inverse Limits of Finite Groups

Definition and Universal Property

Definition. An inverse system of finite groups consists of a directed set $I$, a finite group $G_i$ for each $i \in I$, and a homomorphism $\varphi_{ij} : G_j \to G_i$ for each $i \leq j$, such that $\varphi_{ii} = \mathrm{id}$ and $\varphi_{ij} \circ \varphi_{jk} = \varphi_{ik}$ for $i \leq j \leq k$. Its inverse limit is

$$ \varprojlim_i G_i = \Bigl\{(g_i) \in \prod_i G_i : \varphi_{ij}(g_j) = g_i \text{ for all } i \leq j\Bigr\}, $$

with the subspace topology inherited from the product of the discrete topologies, and with the group structure of the product.

Proposition (universal property). The projections $\pi_i : \varprojlim_j G_j \to G_i$ are continuous homomorphisms satisfying $\varphi_{ij}\pi_j = \pi_i$, and if $H$ is a topological group with compatible continuous homomorphisms $\psi_i : H \to G_i$, then there is a unique continuous homomorphism $\psi : H \to \varprojlim_i G_i$ with $\pi_i \psi = \psi_i$. The inverse limit is therefore the limit of the system in the category of topological groups.

Proof. The subset defined by the compatible families is a subgroup and is closed, being the intersection of the closed conditions $\varphi_{ij}\pi_j = \pi_i$. The map $\psi$ is given by $\psi(h) = (\psi_i(h))_i$; it is a homomorphism, and it is continuous because each coordinate is. Uniqueness is clear.

Theorem. A group $G$ is profinite — isomorphic as a topological group to the inverse limit of an inverse system of finite groups — if and only if $G$ is a compact Hausdorff totally disconnected topological group.

Proof. If $G = \varprojlim G_i$ then $G$ is a closed subgroup of the product of the finite discrete groups $G_i$, hence compact and Hausdorff by Tychonoff, and totally disconnected because each projection to a discrete group has connected fibres and a connected subset of $G$ projects to a point in every $G_i$. Conversely, let $G$ be compact Hausdorff and totally disconnected. The open normal subgroups $N$ form an inverse system (directed by reverse inclusion) and each $G/N$ is finite, because the cosets of an open subgroup cover the compact group $G$ and finitely many suffice. The natural map

$$ G \longrightarrow \varprojlim_{N} G/N $$

is a continuous homomorphism; it is injective because the open subgroups separate points of a totally disconnected compact group (given $x \neq e$, a clopen neighbourhood of $e$ missing $x$ contains an open subgroup by compactness of the complement of a suitable open neighbourhood), and it is surjective by the compatibility of the family of cosets and compactness; a continuous bijection from a compact space to a Hausdorff space is a homeomorphism.

Example. $\mathbb{Z}_p = \varprojlim_k \mathbb{Z}/p^k\mathbb{Z}$ with the maps $\mathbb{Z}/p^{k+1} \to \mathbb{Z}/p^k$ of reduction modulo $p^k$; the element $1$ has for coordinates the classes $1, 1, 1, \dots$, and the group is compact, Hausdorff and totally disconnected. The profinite completion of $\mathbb{Z}$ is

$$ \hat{\mathbb{Z}} = \varprojlim_n \mathbb{Z}/n\mathbb{Z} \cong \prod_{p} \mathbb{Z}_p, $$

the product over all primes, by the Chinese remainder theorem: the inverse system over the divisibility order of $\mathbb{N}$ decomposes into the independent $p$-primary systems.

Profinite Groups as Compact Totally Disconnected Groups

Proposition. The class of profinite groups is closed under taking closed subgroups, quotients by closed normal subgroups, finite products and arbitrary products, and inverse limits. Moreover a continuous surjective homomorphism of profinite groups admits a continuous section (a continuous map, not in general a homomorphism).

Proof. A closed subgroup of a profinite group is compact Hausdorff and totally disconnected, hence profinite. A quotient $G/N$ by a closed normal subgroup is compact Hausdorff, and it is totally disconnected because the quotient map is open and the preimage of a connected subset is a union of cosets; alternatively $G/N$ is the inverse limit of the finite quotients $G/M$ with $M$ open normal containing $N$. Products are compact Hausdorff, and a product of totally disconnected spaces is totally disconnected. For the section: the underlying space of a profinite group is compact Hausdorff and totally disconnected, and a continuous surjection of such spaces admits a continuous section, because the compact Hausdorff totally disconnected spaces are the projective objects among compact Hausdorff spaces (Gleason's theorem, quoted as standard here); the section so obtained is a map of spaces and need not be a homomorphism.

Remark. The section statement is special to profinite groups and is the reason many arguments about them proceed by lifting elements through quotients. It fails for general compact groups: the two-to-one quotient $SU(2) \to SO(3)$ has no continuous section at all, for a section $s$ would make $(U,\varepsilon) \mapsto s(U)\varepsilon$ a homeomorphism $SO(3) \times \{\pm 1\} \to SU(2)$, giving $\pi_1(SU(2)) = 0$ equal to $\pi_1(SO(3) \times \{\pm 1\}) = \mathbb{Z}/2\mathbb{Z}$.

The Topology of a Profinite Group

Open Subgroups and the Neighbourhood Base

Theorem. Let $G$ be a profinite group. Then:

(a) the open subgroups of $G$ are exactly the closed subgroups of finite index, and they form a neighbourhood base at the identity;

(b) every open subgroup contains an open normal subgroup, and an open subgroup of index $n$ contains an open normal subgroup of index dividing $n!$;

(c) the intersection of all open subgroups is $\{e\}$, so the open subgroups separate points;

(d) a subgroup $H$ is open if and only if it is closed and of finite index, and a subgroup of finite index is open if and only if it is closed;

(e) every neighbourhood of the identity contains an open subgroup, and the topology is generated by the open subgroups in the sense that the sets $gN$, $N$ open normal, form a base.

Proof. (a) An open subgroup has finite index by compactness, and it is closed because its complement is a union of cosets, each open. Conversely a closed finite-index subgroup is the complement of finitely many closed cosets and is therefore open. The open subgroups form a neighbourhood base because $G$ is the inverse limit of its finite quotients $G/N$ and the preimage of a point in a finite quotient is a coset of the open subgroup $N$. (b) If $H$ is open then the action of $G$ on $G/H$ is continuous, and the kernel of the action is the intersection of the conjugates of $H$, an open normal subgroup contained in $H$; the index divides $n!$ because the action embeds $G/\ker$ into the symmetric group on $n$ letters. (c) An element $x \neq e$ is separated from $e$ by a clopen set, and since the open subgroups form a neighbourhood base at $e$ by (a), some open subgroup contains $e$ and misses $x$. (d) is immediate from (a): an open subgroup has finite index and is closed; a closed subgroup of finite index is the complement of the finite union of its other cosets, each of which is closed, and is therefore open. (e) restates the definition of the inverse limit topology.

Corollary. A profinite group is recovered from its open normal subgroups: the natural map $G \to \varprojlim_N G/N$, the limit taken over the open normal subgroups $N$, is an isomorphism of topological groups. Consequently the topology of a profinite group is determined by its finite quotients.

Proof. This is the second half of the equivalence of the previous section: the map is a continuous bijection between compact Hausdorff groups, hence a homeomorphism, and it shows that a profinite group can be recovered from the lattice of its open normal subgroups together with the quotient maps.

When the Topology Comes from a Distance

A profinite group is metrisable exactly when its inverse system can be taken countable, and this is the precise sense in which the Krull topology of a large Galois group admits no invariant distance.

Theorem. For a profinite group $G$ the following are equivalent:

(a) $G$ is metrisable;

(b) $G$ is second countable;

(c) $G$ has countably many open subgroups;

(d) $G$ is the inverse limit of a countable inverse system of finite groups.

Proof. A metrisable compact space is second countable, and a second-countable group has countably many open subgroups because each open subgroup is a union of basic open sets and there are countably many of them, with each open subgroup containing a basic open neighbourhood of the identity and hence determined by a subset of a countable base. If there are countably many open subgroups then there are countably many open normal subgroups, whose quotients form a countable inverse system; the map $G \to \varprojlim G/N$ is an isomorphism by the corollary, and the inverse limit of a countable system of finite discrete groups embeds in a countable product of finite sets, which is second countable and metrisable. Conversely a countable inverse limit is metrisable in the topology induced by the metric $d((g_i), (h_i)) = 2^{-\min\{i : g_i \neq h_i\}}$.

Example (a topology that no distance induces). Let $I$ be an uncountable set and let

$$ G = \prod_{i \in I} \mathbb{Z}/2\mathbb{Z}, $$

with the product topology. Then $G$ is compact, Hausdorff and totally disconnected, hence profinite; its open subgroups are the preimages of the subgroups of the finite subproducts $\prod_{i \in F} \mathbb{Z}/2\mathbb{Z}$ with $F$ finite; there are $|I|$ of them, since there are $|I|$ finite subsets $F$ and only finitely many subgroups of each finite subproduct, while the kernels of the coordinate projections already give $|I|$ distinct open subgroups. Hence there are uncountably many open subgroups, the group is not second countable, and it carries no metric inducing its topology: the Krull topology of a large Galois group is not induced by any distance on the group. The same argument applies to $\mathbb{Z}_p^{\,I}$ for uncountable $I$ and to the absolute Galois group of a field with uncountably many finite Galois extensions.

Remark. Metrisability is nevertheless the common case in arithmetic: the absolute Galois group $\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ of the rationals has countably many open subgroups — the finite extensions of $\mathbb{Q}$ are countable — and is therefore second countable and metrisable, even though it is not finitely generated as a profinite group; its abelianisation $\hat{\mathbb{Z}}^\times = \prod_p \mathbb{Z}_p^\times$ is not finitely generated. The topological statements of Galois theory are usually made for a general, possibly non-metrisable, extension, and the distinction is precisely the countability of the lattice of open subgroups.

The Krull Topology

The Topology on a Galois Group

Let $L/K$ be a Galois extension, possibly infinite, and let $\operatorname{Gal}(L/K)$ be its group of $K$-automorphisms. For each finite Galois subextension $M/K$ with $K \subseteq M \subseteq L$, restriction gives a homomorphism

$$ \operatorname{Gal}(L/K) \longrightarrow \operatorname{Gal}(M/K), $$

and the finite Galois subextensions form a directed set under inclusion.

Definition. The Krull topology on $\operatorname{Gal}(L/K)$ is the coarsest topology for which all the restriction maps $\operatorname{Gal}(L/K) \to \operatorname{Gal}(M/K)$ to finite Galois subextensions are continuous, the finite groups $\operatorname{Gal}(M/K)$ carrying the discrete topology. A basis of neighbourhoods of the identity is the family

$$ \operatorname{Gal}(L/M) = \{\sigma \in \operatorname{Gal}(L/K) : \sigma|_M = \mathrm{id}_M\}, \qquad M/K \text{ finite Galois}. $$

Theorem. With the Krull topology, $\operatorname{Gal}(L/K)$ is a profinite group:

$$ \operatorname{Gal}(L/K) \cong \varprojlim_{M/K \text{ finite Galois}} \operatorname{Gal}(M/K), $$

the isomorphism carrying $\sigma$ to the family $(\sigma|_M)_M$. Consequently $\operatorname{Gal}(L/K)$ is compact, Hausdorff and totally disconnected, and its open subgroups are the subgroups $\operatorname{Gal}(L/M)$ for finite Galois $M/K$.

Proof. The restriction maps are compatible: if $M \subseteq M'$ then the restriction $\operatorname{Gal}(M'/K) \to \operatorname{Gal}(M/K)$ composes correctly, so there is a homomorphism to the inverse limit. It is injective because an automorphism of $L$ fixing every finite Galois subextension fixes $L$, every element of $L$ lying in such a subextension. It is surjective because a compatible family of automorphisms of the finite Galois subextensions extends to a $K$-automorphism of the union $L$: the family is consistent on overlaps by compatibility, and defines an automorphism of the union. The map is continuous with continuous inverse by the definition of the Krull topology as the inverse limit topology. The description of the open subgroups is the theorem on open subgroups applied to the inverse system.

The Fundamental Theorem of Galois Theory

The point of the topology is that it makes the Galois correspondence exact.

Theorem (Krull). Let $L/K$ be a Galois extension with group $G = \operatorname{Gal}(L/K)$ carrying the Krull topology. The assignments

$$ M \longmapsto \operatorname{Gal}(L/M), \qquad H \longmapsto L^H, $$

are mutually inverse bijections between the intermediate fields $K \subseteq M \subseteq L$ and the closed subgroups $H \subseteq G$. Under the bijection:

(a) $M/K$ is finite if and only if $\operatorname{Gal}(L/M)$ is open, and then $[M : K] = [G : \operatorname{Gal}(L/M)]$;

(b) $M/K$ is Galois if and only if $\operatorname{Gal}(L/M)$ is a closed normal subgroup, and then $\operatorname{Gal}(M/K) \cong G/\operatorname{Gal}(L/M)$;

(c) the finite subextensions correspond exactly to the open subgroups, and the normal open subgroups correspond to the finite Galois subextensions;

(d) an intermediate field $M$ with $L/M$ Galois corresponds to a closed subgroup, and $M = L^H$ for $H = \operatorname{Gal}(L/M)$.

Proof. The standard argument of Galois Theory establishes the bijection between intermediate fields and subgroups of $G$ in the finite case, and the infinite case is the statement that the correspondence is continuous for the Krull topology: the fixed field of a subgroup $H$ equals the fixed field of its closure, because an element fixed by $H$ is fixed by the closure of $H$ acting continuously on $L$ carrying the discrete topology on its elements, so the bijection is between fields and closed subgroups. The index and normality statements are the finite Galois theory applied to each finite Galois subextension, together with the description of the open subgroups above: $\operatorname{Gal}(L/M)$ is open exactly when it contains $\operatorname{Gal}(L/M')$ for some finite Galois $M'$, which is exactly the statement that $M$ is contained in a finite Galois extension $M'$ of $K$, and then $[M : K] \leq [M' : K]$ is finite.

Example. For $L = \bar{\mathbb{F}}_p$ and $K = \mathbb{F}_p$, the finite Galois subextensions are the fields $\mathbb{F}_{p^n}$, and the Frobenius automorphism $\mathrm{Fr} : x \mapsto x^p$ has image generating each $\operatorname{Gal}(\mathbb{F}_{p^n}/\mathbb{F}_p) \cong \mathbb{Z}/n\mathbb{Z}$. Hence

$$ \operatorname{Gal}(\bar{\mathbb{F}}_p/\mathbb{F}_p) \cong \varprojlim_n \mathbb{Z}/n\mathbb{Z} = \hat{\mathbb{Z}} = \prod_q \mathbb{Z}_q, $$

topologically generated by the Frobenius, and its closed subgroups are the subgroups $m\hat{\mathbb{Z}}$, each of index $m$ and corresponding to the finite extension $\mathbb{F}_{p^m}$; every closed subgroup of $\hat{\mathbb{Z}}$ has finite index and is therefore open.

Example. For $K = \mathbb{Q}$ and $L = \mathbb{Q}(\zeta_{p^\infty}) = \bigcup_k \mathbb{Q}(\zeta_{p^k})$, the Galois group is $\mathbb{Z}_p^\times$: a $p$-adic unit acts on the compatible system of $p^k$-th roots of unity, and the resulting map $\mathbb{Z}_p^\times \to \varprojlim_k (\mathbb{Z}/p^k\mathbb{Z})^\times$ is an isomorphism by the compatibility. The Krull topology here is the $p$-adic topology of the units, and the closed subgroups of $\mathbb{Z}_p^\times$ correspond to the subfields of $\mathbb{Q}(\zeta_{p^\infty})$. The structure of the units is the standard one: for odd $p$, $\mathbb{Z}_p^\times \cong \mathbb{Z}/(p-1)\mathbb{Z} \times \mathbb{Z}_p$, while $\mathbb{Z}_2^\times \cong \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}_2$, isomorphisms that are continuous for the profinite and $p$-adic topologies and that determine the closed subgroup lattice of the Galois group.

The Profinite Completion

Definition and Functoriality

Definition. Let $G$ be an abstract group. Its profinite completion is

$$ \hat G^{\mathrm{pf}} = \varprojlim_{N \trianglelefteq G,\ [G:N] < \infty} G/N, $$

the inverse limit over the finite-index normal subgroups with the natural map $\iota_G : G \to \hat G^{\mathrm{pf}}$. The hat without the superscript is reserved in this corpus for the completion of a topological group in its two-sided uniformity, as in Topological Groups; the two are related but distinct, and the superscript is used whenever ambiguity is possible.

Theorem. The profinite completion $\hat G^{\mathrm{pf}}$ is a profinite group and $\iota_G$ has dense image. The assignment $G \mapsto \hat G^{\mathrm{pf}}$ is a functor from groups to profinite groups, left adjoint to the forgetful functor: for a profinite group $P$,

$$ \operatorname{Hom}_{\mathrm{prof}}(\hat G^{\mathrm{pf}}, P) \cong \operatorname{Hom}_{\mathrm{grp}}(G, P). $$

The map $\iota_G$ is injective precisely when $G$ is residually finite.

Proof. Density: a basic open neighbourhood of an element of the limit fixes finitely many coordinates, and compatibility plus the Chinese-remainder argument produces an element of the image of $G$ in that neighbourhood. Adjointness: a homomorphism $f : G \to P$ with $P$ profinite pulls back each open normal subgroup $M \trianglelefteq_o P$ to a finite-index normal subgroup $f^{-1}(M) \trianglelefteq G$, and it induces $G/f^{-1}(M) \to P/M$; the subgroups $f^{-1}(M)$ are cofinal in the inverse system defining $\hat G^{\mathrm{pf}}$, so these compatible maps assemble into a continuous homomorphism $\hat G^{\mathrm{pf}} \to \varprojlim_M P/M = P$, unique because it agrees with $f$ on the dense image of $G$. Injectivity of $\iota_G$ is by definition of residual finiteness: the kernel is the intersection of the finite-index normal subgroups, which is trivial exactly when every nontrivial element survives in a finite quotient.

Example. $\hat{\mathbb{Z}} = \prod_p \mathbb{Z}_p$; the completion of a finite group is itself; the completion of a free group $F_n$ is the free profinite group $\hat F_n$, and $\iota$ is injective because free groups are residually finite. The completion of $GL_n(\mathbb{Z})$ is $\varprojlim_m GL_n(\mathbb{Z}/m\mathbb{Z})$, described by the congruence subgroups. Finitely generated abelian groups and finitely generated linear groups are residually finite, so their completions contain them densely.

Pro-p Groups and the Structure Theory

Definition. A pro-$p$ group is a profinite group whose open subgroups are $p$-groups, equivalently a projective limit of finite $p$-groups; a Sylow $p$-subgroup of a profinite group $G$ is a maximal pro-$p$ closed subgroup.

Theorem (Sylow theory for profinite groups). Every profinite group contains a Sylow $p$-subgroup for every prime $p$, and any two Sylow $p$-subgroups are conjugate.

Proof. A finite quotient of $G$ has a Sylow $p$-subgroup, and the Sylow subgroups of the finite quotients are compatible after conjugation; passing to a compatible family by compactness produces a maximal pro-$p$ closed subgroup. Conjugacy is the standard argument, using the corresponding finite statements and compactness.

Theorem (the Frattini subgroup). In a pro-$p$ group $G$ every maximal closed subgroup is open of index $p$, and the intersection of the maximal open subgroups is the Frattini subgroup $\Phi(G)$; the quotient $G/\Phi(G)$ is an elementary abelian pro-$p$ group.

Proof. Let $M$ be a maximal closed subgroup of the pro-$p$ group $G$ and let $N$ be an open normal subgroup. The subgroup $MN$ is open and contains $M$, so maximality gives $MN = M$ or $MN = G$. If $MN = G$ then $[G : M] = [N : M \cap N] \leq [G : N]$ is finite; if $MN = M$ then $N \subseteq M$. If $N \subseteq M$ for every open normal subgroup $N$, then $M$ contains the intersection of the open normal subgroups, which is trivial, so $M = \{e\}$ and $G$ has no proper nontrivial closed subgroup; then every finite quotient of $G$ is $\mathbb{Z}/p\mathbb{Z}$ and $G$ is procyclic, hence $\mathbb{Z}/p\mathbb{Z}$, since a procyclic pro-$p$ group is $\mathbb{Z}/p^n\mathbb{Z}$ or $\mathbb{Z}_p$ and $\mathbb{Z}_p$ has the proper closed subgroup $p\mathbb{Z}_p$. In every case $[G : M]$ is finite, hence a power of $p$, so $M$ is open; the finite $p$-group $G/M$ has no proper subgroup, so it is $\mathbb{Z}/p\mathbb{Z}$ and $M$ is normal of index $p$. The Frattini subgroup, the intersection of the maximal open subgroups, is closed and normal, and for a pro-$p$ group it satisfies $\Phi(G) = \overline{G^p[G,G]}$, so that $G/\Phi(G)$ is elementary abelian.

Theorem (generation). Let $G$ be a pro-$p$ group and let $S \subseteq G$. Then $S$ generates $G$ as a topological group if and only if the image of $S$ generates the elementary abelian pro-$p$ group $G/\Phi(G)$. When $G/\Phi(G)$ is finite of order $p^d$, the integer $d$ is the rank of $G$ and is the minimal number of topological generators of $G$.

Proof. This is the profinite form of the Burnside basis theorem. Suppose first that $S$ does not generate $G$ and put $H = \overline{\langle S\rangle}$, a proper closed subgroup. Since $H$ is closed and proper, some coset $gN$ of an open normal subgroup $N$ misses $H$; then $HN \neq G$, and $HN$ is an open subgroup containing $H$, so it is contained in a maximal open subgroup $M$, which is the preimage of a maximal subgroup of a finite quotient. Now $M/\Phi(G)$ is a proper subgroup of $G/\Phi(G)$, because $M \neq G$ and $\Phi(G) \subseteq M$; since $S \subseteq M$, the image of $S$ does not generate $G/\Phi(G)$. Conversely, if the image of $S$ generates $G/\Phi(G)$ and $H = \overline{\langle S\rangle}$ were proper, the argument would produce a maximal open $M \supseteq H$, and the image of $S$ would lie in the proper subgroup $M/\Phi(G)$, a contradiction. For the rank: if $G$ is generated by $d$ elements then so is $G/\Phi(G)$, so $|G/\Phi(G)| \leq p^d$; conversely a generating set of the finite elementary abelian group $G/\Phi(G)$ lifts to a topological generating set of $G$ by the first part.

Example. $\mathbb{Z}_p$ is a pro-$p$ group of rank $1$, with $\Phi(\mathbb{Z}_p) = p\mathbb{Z}_p$ and $\mathbb{Z}_p/\Phi(\mathbb{Z}_p) \cong \mathbb{Z}/p\mathbb{Z}$; it is not a finite $p$-group, being uncountable. The free pro-$p$ group on $d$ generators has rank $d$. In the profinite abelian group $\hat{\mathbb{Z}} = \prod_q \mathbb{Z}_q$ the Sylow $p$-subgroup is the factor $\mathbb{Z}_p$, the maximal open subgroups are the kernels of the projections $\hat{\mathbb{Z}} \to \mathbb{Z}/q\mathbb{Z}$, and $\Phi(\hat{\mathbb{Z}}) = \prod_q q\mathbb{Z}_q$.

Summary

A profinite group is an inverse limit of finite groups, equivalently a compact Hausdorff totally disconnected topological group; the class is closed under closed subgroups, quotients by closed normal subgroups, products and inverse limits, and a surjection of profinite groups has a continuous section. Its open subgroups are exactly its closed finite-index subgroups and form a neighbourhood base at the identity; every open subgroup contains an open normal subgroup, and a profinite group is recovered from its finite quotients. The topology is metrisable exactly when there are countably many open subgroups, so it is induced by no distance for $\prod_{i \in I} \mathbb{Z}/2\mathbb{Z}$ with $I$ uncountable; the absolute Galois group of a field with uncountably many finite Galois extensions is the standard arithmetic instance.

The Krull topology on a Galois group is the inverse-limit topology over the finite Galois subextensions, and it makes the fundamental theorem of Galois theory a bijection between intermediate fields and closed subgroups; open subgroups correspond to finite subextensions and open normal subgroups to finite Galois subextensions. The principal arithmetic examples are $\operatorname{Gal}(\bar{\mathbb{F}}_p/\mathbb{F}_p) \cong \hat{\mathbb{Z}}$ and $\operatorname{Gal}(\mathbb{Q}(\zeta_{p^\infty})/\mathbb{Q}) \cong \mathbb{Z}_p^\times$. Every abstract group has a profinite completion, left adjoint to the forgetful functor and injective exactly for residually finite groups; for $\mathbb{Z}$ it is $\hat{\mathbb{Z}} = \prod_p \mathbb{Z}_p$. Pro-$p$ groups carry Sylow theory, the Frattini subgroup and the rank $d = \dim G/\Phi(G)$ governing topological generation. Integration over a profinite group and its $L^p$ theory are Part III.

Summary of Notation

Symbol	Meaning
$\varprojlim_i G_i$	Inverse limit of an inverse system of groups
$\varphi_{ij} : G_j \to G_i$	Bonding maps of the system
profinite	Inverse limit of finite groups; compact Hausdorff totally disconnected
$N \trianglelefteq_o G$	Open (hence finite-index) normal subgroup
$G/N$	Finite quotient, discrete
$\hat G^{\mathrm{pf}}$	Profinite completion of an abstract group $G$
$\mathbb{Z}_p$, $\hat{\mathbb{Z}}$	$p$-adic integers; $\hat{\mathbb{Z}} = \prod_p \mathbb{Z}_p$
$\operatorname{Gal}(L/K)$	Galois group of an extension, with the Krull topology
$\operatorname{Gal}(L/M)$, $M/K$ finite Galois	Basic neighbourhood of the identity in the Krull topology
$L^H$	Fixed field of a closed subgroup $H$
$\mathrm{Fr}$	Frobenius automorphism, topological generator of $\operatorname{Gal}(\bar{\mathbb{F}}_p/\mathbb{F}_p)$
pro-$p$ group	Projective limit of finite $p$-groups
$\Phi(G)$	Frattini subgroup: intersection of the maximal open subgroups
$d = \dim_{\mathbb{F}_p} G/\Phi(G)$	Rank; minimal number of topological generators
Sylow $p$-subgroup	Maximal pro-$p$ closed subgroup
residually finite	Every nontrivial element survives in a finite quotient
$D \mapsto D^\vee$	Pontryagin dual of a discrete torsion group, a profinite abelian group; used for the abelian profinite case