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Profinite group

A profinite group is a topological group that arises as the inverse limit of an inverse system of finite discrete groups.^[1] Equivalently, it is a compact, Hausdorff, and totally disconnected topological group.^[2] This structure endows profinite groups with a rich topology where open subgroups have finite index, and the identity element has a basis of neighborhoods consisting of normal open subgroups.^[3] Profinite groups generalize finite groups in a topological setting, preserving many algebraic properties such as the existence of Sylow subgroups for pro-p variants and the validity of certain finiteness theorems.^[1] Key examples include the p-adic integers \mathbb{Z}_p, which form the inverse limit of the rings \mathbb{Z}/p^n\mathbb{Z}, and the profinite completion \hat{\mathbb{Z}} of the integers, isomorphic to the product \prod_p \mathbb{Z}_p over all primes p.^[2] Closed subgroups and continuous quotients of profinite groups remain profinite, ensuring closure under standard group operations.^[3] In number theory and algebraic geometry, profinite groups play a central role, particularly as absolute Galois groups of fields, which are profinite and describe infinite Galois extensions via their finite quotients.^[1] For instance, the absolute Galois group \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) is profinite, and its maximal abelian quotient is isomorphic to the profinite completion of the units in the cyclotomic fields.^[2] They also appear in the study of étale fundamental groups of algebraic varieties, where the profinite completion captures the action on finite covers.^[3]

Definitions

Constructive definition

A profinite group can be constructively defined as the inverse limit of an inverse system of finite discrete groups connected by surjective homomorphisms.^[4] This approach emphasizes the approximation of the group by its finite quotients, capturing infinite structures through compatible finite data.^[3] To formalize this, consider an inverse system indexed by a directed partially ordered set I, consisting of finite groups \{G_i\}_{i \in I} (each equipped with the discrete topology) and surjective bonding maps f_{ij}: G_j \to G_i for all i \leq j in I, satisfying the conditions f_{ii} = \mathrm{id}_{G_i} and f_{ik} = f_{ij} \circ f_{jk} whenever i \leq j \leq k.^[4] A directed set I ensures that for any pair i, j \in I, there exists k \in I such that i \leq k and j \leq k, allowing the system to "funnel" towards a limit.^[1] The inverse limit, denoted \varprojlim_{i \in I} G_i, is the subset of the direct product \prod_{i \in I} G_i consisting of all threads (g_i)_{i \in I} such that f_{ij}(g_j) = g_i for all i \leq j; this set forms a group under componentwise multiplication.^[3] The universal property of the inverse limit characterizes profinite groups algebraically: for any group H and a compatible family of homomorphisms \{\phi_i: H \to G_i\}_{i \in I} (meaning \phi_i = f_{ij} \circ \phi_j for i \leq j), there exists a unique homomorphism \phi: H \to \varprojlim G_i such that the composition with each projection \pi_i: \varprojlim G_i \to G_i yields \phi_i.^[4] This property ensures that the inverse limit is the "universal" object encoding all compatible maps to the finite groups in the system.^[1] The topology on the profinite group \varprojlim G_i is induced as the subspace topology from the product topology on \prod G_i, where each finite discrete space G_i is compact and Hausdorff.^[3] This yields a compact, Hausdorff topology on the limit, with a basis of neighborhoods of the identity element consisting of the kernels \ker(\pi_i) = \{ (g_j)_{j \in I} \mid g_i = e_i \} for each i \in I, which are open normal subgroups of finite index.^[4] A key example of this construction is the profinite completion of an arbitrary group G, defined as \hat{G} = \varprojlim_{N} G/N, where the inverse system is indexed by the directed set of all normal subgroups N \trianglelefteq G of finite index, ordered by reverse inclusion (so N \leq M if M \subseteq N), with bonding maps the natural surjections G/M \to G/N.^[1] The canonical homomorphism G \to \hat{G} sending g \mapsto (gN)_N has dense image, and \hat{G} satisfies the universal property for maps from G to finite groups factoring through finite quotients.^[3] This completion embeds G into a profinite group that "remembers" all its finite quotients.^[4]

Axiomatic definition

A profinite group is defined axiomatically as a topological group that is compact, Hausdorff, and totally disconnected.^[5] In this setting, compactness ensures that every open cover admits a finite subcover, the Hausdorff property allows separation of distinct points by disjoint open sets, and total disconnectedness means that the connected component of the identity element is trivial.^[5] Central to this axiomatization is the role of open normal subgroups in generating the topology. Specifically, every profinite group possesses a basis of neighborhoods at the identity consisting of open normal subgroups of finite index.^[5] This basis property implies that every neighborhood of the identity contains such a normal open subgroup, enabling the topology to be determined by the collection of these subgroups ordered by inclusion.^[5] An alternative axiomatization characterizes a profinite group as a topological group that arises as the inverse limit of its finite quotients, though this formulation aligns equivalently with the topological axioms above.^[5]

Equivalence of definitions

The constructive definition of a profinite group as an inverse limit of finite discrete groups is equivalent to the axiomatic definition as a compact, Hausdorff, totally disconnected topological group. This equivalence establishes a unified framework for the theory, relying on the topological residual finiteness of such groups, where the intersection of all open normal subgroups of finite index is the trivial subgroup. To see that every inverse limit G = \lim_{\leftarrow} G_i of finite discrete groups satisfies the axiomatic properties, equip G with the subspace topology from the product \prod_i G_i, where each G_i has the discrete topology. Compactness follows from Tychonoff's theorem, as each finite discrete space is compact, their product is compact, and the inverse limit is a closed subset thereof.^[2] Hausdorffness holds because the product topology separates points, and closed subsets inherit this property.^[2] Total disconnectedness arises from the existence of a basis of clopen neighborhoods at the identity given by the kernels of the projection maps \pi_i: G \to G_i, which separate points via cosets in the finite quotients.^[3] Conversely, every compact Hausdorff totally disconnected topological group G is isomorphic as a topological group to an inverse limit of its finite continuous quotients. In such a G, the open subgroups form a basis of neighborhoods at the identity, and compactness ensures that every open subgroup has finite index; moreover, there exists a basis of open normal subgroups of finite index.^[2] The canonical inverse system is formed by the quotients G/N, where N ranges over the directed set of open normal subgroups of finite index, ordered by reverse inclusion, with transition maps the natural projections.^[3] The canonical map \phi: G \to \lim_{\leftarrow} G/N sending g \mapsto (gN)_N is continuous, surjective, and injective because G is topologically residually finite (the kernels separate points due to Hausdorffness), hence a homeomorphism onto its image, which is the full inverse limit by density and compactness.^[2]

Profinite completion

The profinite completion of a group G, denoted \hat{G}, is defined as the inverse limit \lim_{\leftarrow N} G/N, where the limit is taken over all normal subgroups N \trianglelefteq G of finite index, ordered by reverse inclusion, and equipped with the quotient maps G/N \to G/M for M \subseteq N.^[6] There is a natural homomorphism \iota: G \to \hat{G} given by \iota(g) = (gN)_{N}, whose image is dense in \hat{G}.^[6] This construction embeds G into a profinite group that captures all its finite quotients.^[3] The profinite completion satisfies a universal property: for any profinite group H and any group homomorphism \phi: G \to H, there exists a unique continuous homomorphism \hat{\phi}: \hat{G} \to H such that \phi = \hat{\phi} \circ \iota.^[6] This makes \hat{G} the universal profinite quotient of G, in the sense that every homomorphism from G to a profinite group factors uniquely through \hat{G}.^[3] The kernel of the natural map \iota: G \to \hat{G} is the intersection of all finite-index normal subgroups of G, known as the profinite radical of G.^[6] If G is already profinite, then \hat{G} \cong G via \iota, which is an isomorphism (idempotence of the completion functor).^[6] For example, the profinite completion of a free group F on n generators is the free profinite group on n generators.^[6]

Examples

Finite groups as profinite

Every finite group, when equipped with its discrete topology, is a profinite group. This follows because such a group G can be viewed as the inverse limit of the trivial inverse system consisting solely of itself, where the transition maps are the identity.^[7] The discrete topology on a finite set ensures compactness, as every finite topological space is compact, and total disconnectedness, since singletons are both open and closed.^[8] Finite profinite groups are precisely those profinite groups possessing only finitely many open subgroups. In a finite discrete profinite group, all subgroups are open due to the discrete topology, and there are only finitely many subgroups in total. Conversely, if a profinite group has finitely many open subgroups, their finite intersection yields the trivial subgroup as open, implying a discrete topology; combined with the defining compactness of profinite groups, this forces the group to be finite.^[8] For any finite group G, its profinite completion coincides with G itself, endowed with the discrete topology, as the natural map G \to \hat{G} is an isomorphism.^[7] A concrete example is the symmetric group S_n on n letters, which is finite of order n! and thus profinite under the discrete topology; it serves as a basic illustration of permutation groups within the profinite framework.^[8]

Infinite profinite groups

A canonical example of an infinite profinite group is the additive group of p-adic integers \mathbb{Z}_p for a fixed prime p, constructed as the inverse limit \varprojlim_n \mathbb{Z}/p^n \mathbb{Z} equipped with the p-adic topology, in which addition and multiplication are continuous operations.^[2] This group is compact, totally disconnected, and Hausdorff, with the subgroups p^n \mathbb{Z}_p forming a basis of open neighborhoods of the identity.^[2] Another class of infinite profinite groups arises as infinite products of finite groups, endowed with the product topology. For instance, the direct product \prod_{p \text{ prime}} \mathbb{Z}/p\mathbb{Z} over all primes p is profinite, as it is an inverse limit of finite quotients and compact by Tychonoff's theorem.^[2] In general, arbitrary products of profinite groups, such as \prod_{i \in I} \mathbb{Z}/2\mathbb{Z} for an infinite index set I, yield profinite groups via the product topology.^[2] Infinite profinite groups are always uncountable, as any countable compact Hausdorff space that is totally disconnected must be finite.^[3] Even compact metric profinite groups, such as \mathbb{Z}_p, are uncountable despite admitting a countable basis for their topology.^[3] Profinite components also appear in the adele ring of the rationals \mathbb{Q}, where the finite adele ring includes the restricted direct product \prod_p' \mathbb{Z}_p over primes p, with respect to the compact open subgroups \mathbb{Z}_p; this coincides with the full product \prod_p \mathbb{Z}_p and forms a profinite ring under the restricted product topology.^[2] Unlike finite groups, which carry the discrete topology, these infinite profinite groups are compact but non-discrete, with no isolated points.^[2]

Absolute Galois groups

The absolute Galois group of a field K, denoted \mathrm{Gal}(\overline{K}/K) where \overline{K} is a separable algebraic closure of K, is the group of automorphisms of \overline{K} fixing K. It is realized as the inverse limit \varprojlim \mathrm{Gal}(L/K) taken over all finite Galois extensions L/K of K, ordered by inclusion, and equipped with the Krull topology in which a basis of neighborhoods of the identity consists of the open normal subgroups \mathrm{Gal}(\overline{K}/L) for finite Galois L/K. This topological group structure renders \mathrm{Gal}(\overline{K}/K) profinite.^[9] A prominent example is \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}), the absolute Galois group of the rationals, which is profinite but non-abelian since it surjects onto non-abelian finite groups such as S_3. The Krull topology on this group ensures that closed subgroups correspond bijectively to subextensions of \overline{\mathbb{Q}}/\mathbb{Q}, with finite-index closed subgroups corresponding to finite extensions.^[9] The maximal pro-p quotient of \mathrm{Gal}(\overline{K}/K), for a prime p, is the Galois group of the maximal pro-p extension of K, obtained as the inverse limit of all finite [p-group](/page/P-group) quotients; this quotient captures the p-adic extensions of K in the sense that its continuous representations classify pro-p Galois extensions.^[10] For abelian cases, such as the local field K = \mathbb{Q}_p, local class field theory establishes that the abelianization \mathrm{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)^{\mathrm{ab}} is topologically isomorphic to the profinite completion \widehat{\mathbb{Q}_p^\times} via the Artin reciprocity map, where \mathbb{Q}_p^\times \cong \mathbb{Z} \times \mathbb{Z}_p^\times \times \mu_{p-1} (for p odd) highlights the role of the p-adic units \mathbb{Z}_p^\times in the ramified part. Pontryagin duality for these locally compact abelian groups underpins the isomorphism, ensuring compatibility with the topological structures.^[11] Absolute Galois groups are residually finite, meaning the intersection of all finite-index normal subgroups is trivial, and their finite quotients classify the finite Galois extensions of K via the fundamental theorem of infinite Galois theory.^[9]

Topological properties

Compactness and totally disconnectedness

A profinite group G, defined as the inverse limit of finite discrete groups, inherits compactness from its construction. Specifically, G embeds as a closed subgroup in the product \prod_i G_i of the finite groups G_i, each equipped with the discrete topology, making the product compact by Tychonoff's theorem.^[2]^[12] As a closed subset of a compact space, G itself is compact in the subspace topology.^[2] Profinite groups are also totally disconnected, meaning that every connected component consists of a single point and there are no non-trivial connected subsets. This property arises because the topology on G admits a basis of clopen (both open and closed) sets, generated by the kernels of the continuous surjections to the finite quotients G_i.^[12]^[13] For any two distinct points x, y \in G, there exists a clopen neighborhood of x excluding y, ensuring separation without connected continua.^[2] The presence of a basis of clopen sets further implies that profinite groups are zero-dimensional topological spaces, where dimension is measured by the existence of such a basis for the topology.^[13]^[12] In the axiomatic characterization, profinite groups correspond precisely to compact, totally disconnected Hausdorff topological groups, which are equivalent to Stone spaces endowed with a compatible group structure.^[12] Stone spaces themselves are the compact, totally disconnected Hausdorff spaces, mirroring the inverse limit structure of profinite objects.^[12]^[13]

Hausdorff topology and uniformity

Profinite groups are Hausdorff topological groups, with the separation axiom arising directly from their residual finiteness. Specifically, the trivial subgroup \{e\} is closed in the profinite topology, as it equals the intersection of all open normal subgroups of the group, each of which has finite index. This property ensures that distinct elements can be separated by open sets, confirming the Hausdorff condition essential for the topological structure of profinite groups.^[8] The topology on a profinite group G induces a compatible uniform structure that is left-invariant, meaning it is preserved under left translations by elements of G. A basis for this uniformity consists of the entourages U_N = \{(g,h) \in G \times G \mid g h^{-1} \in N\} , where \{N_n\} is a basis of neighborhoods of the identity consisting of open normal subgroups. These entourages capture the "closeness" relation in a way that aligns with the inverse limit construction of profinite groups, ensuring the uniform structure reflects the finite quotients underlying G.^[8] In this uniform structure, the group operations—multiplication and inversion—are uniformly continuous. This follows from the left-invariance of the uniformity and the fact that continuous maps from compact spaces to uniform spaces are uniformly continuous, a property inherent to the compact nature of profinite groups. Moreover, profinite groups are complete as uniform spaces, as every compact uniform space is complete; their compactness thus guarantees the absence of proper uniform completions.^[8]

Continuous homomorphisms

A continuous group homomorphism \phi: G \to H between profinite groups G and H is a group homomorphism that is also continuous as a map of topological spaces. Since the topology on each profinite group admits a basis of neighborhoods of the identity consisting of open normal subgroups, \phi is continuous if and only if \phi^{-1}(U) is open in G for every open normal subgroup U of H.^[3]^[4] Any continuous homomorphism \phi: G \to H between profinite groups is a closed map, as the image of the compact set G is compact and hence closed in the Hausdorff space H. Moreover, if \phi is surjective, then it is an open map; this follows from the compactness of G and the fact that surjective continuous homomorphisms from compact topological groups to Hausdorff topological groups are quotient maps.^[1] Continuous homomorphisms between profinite groups are intimately related to their presentations as inverse limits of finite groups. Specifically, if G = \lim_{\leftarrow} \{G_i\} and H = \lim_{\leftarrow} \{H_j\} are inverse systems of finite groups with bonding maps, then a group homomorphism \phi: G \to H is continuous if and only if it is compatible with the inverse systems, meaning that for every i and j, the induced map G_i \to H_j (whenever defined) respects the bonding maps of the systems. This compatibility ensures that \phi preserves the topological structure induced by the finite quotients.^[3]^[4] The image \phi(G) of a continuous homomorphism \phi: G \to H with G profinite is dense in H if and only if \phi is surjective on every finite quotient, i.e., for every open normal subgroup U of H, the induced map G / \phi^{-1}(U) \to H / U is surjective. Since \phi(G) is compact and thus closed in the Hausdorff space H, density implies that \phi(G) = H, so the homomorphism is surjective.^[3] A canonical example of a continuous homomorphism is the projection \pi_K: G \to G/K from a profinite group G onto a finite quotient G/K, where K is an open normal subgroup of G. This map is a continuous surjection, as K is open and the quotient topology on the finite discrete group G/K coincides with the discrete topology. Such projections form the bonding maps in the inverse system defining G.^[1]^[4]

Algebraic properties

Subgroups and quotients

In profinite groups, open subgroups are precisely those that are closed and of finite index. This equivalence arises because the topology on a profinite group G admits a basis of neighborhoods consisting of open normal subgroups of finite index, ensuring that any finite-index subgroup contains such a basis element and is thus open, while open subgroups, being compact in a compact space, have finite index.^[5] Closed subgroups of a profinite group G are exactly the intersections of open subgroups containing them. Every closed subgroup of G is itself profinite, inheriting the compact, totally disconnected Hausdorff topology from G. A closed subgroup is profinite in particular if it is open or of finite index, though the profiniteness holds more generally for all closed subgroups.^[5] For quotients, if N is a closed normal subgroup of a profinite group G, then the quotient G/N is profinite when equipped with the quotient topology. In the special case where N is open (and thus normal and of finite index), G/N is a finite discrete group, hence profinite. The natural quotient map G \to G/N is continuous. Every closed normal subgroup of G is the intersection of the open normal subgroups containing it.^[5]^[14] Profinite groups admit presentations analogous to discrete groups: every profinite group is isomorphic to a quotient of a free profinite group on a suitable generating set by a closed normal subgroup generated by a set of relations.^[3]

Normal subgroups and extensions

In profinite groups, open normal subgroups play a fundamental role in defining the topology. Specifically, the collection of all open normal subgroups of a profinite group G forms a basis for the neighborhoods of the identity element, and the quotient G/N by any such open normal subgroup N is a finite discrete group.^[3] This property ensures that the topology is determined by these finite quotients, reflecting the inverse limit structure of profinite groups.^[15] Closed normal subgroups preserve the profinite nature of quotients. If N is a closed normal subgroup of a profinite group G, then the quotient G/N, equipped with the quotient topology, is itself profinite.^[3] This follows from the fact that closed subgroups of profinite groups are profinite, and the quotient map is continuous and open.^[15] Short exact sequences capture extensions in the profinite category. Consider a short exact sequence of topological groups

$1 \to N \to G \xrightarrow{\pi} Q \to 1,

where N and Q are profinite, the maps are continuous homomorphisms, and N = \ker \pi is closed normal in G. Under these conditions, G is profinite.^[16] Such sequences arise naturally, for example, in the context of Galois groups where subextensions correspond to closed normal subgroups.^[16] Group extensions in the profinite setting are classified using continuous cohomology. An extension of a profinite group Q by a profinite group N (viewed as a discrete topological Q-module) corresponds to a continuous 2-cocycle, and the equivalence classes of such extensions are in bijection with the second continuous cohomology group H^2(Q, N).^[15] Profinite cohomology, defined using continuous cochains on profinite groups with finite modules, generalizes classical group cohomology to the topological context and is essential for studying these structures.^[15] A notable structural property is that profinite groups contain no non-trivial divisible subgroups. This holds because finite groups, being the building blocks via inverse limits, have no non-trivial divisible subgroups, and the property is preserved under inverse limits.

Finiteness conditions

Profinite groups are inherently residually finite, meaning that the intersection of all normal subgroups of finite index is the trivial subgroup.^[17] This property follows from the topological structure, where open normal subgroups form a basis of neighborhoods of the identity, ensuring that non-identity elements can be separated by homomorphisms onto finite groups.^[3] In particular, for any non-trivial element g \in G, there exists an open normal subgroup N \trianglelefteq G of finite index such that g \notin N.^[17] A special class of profinite groups are the pro-p groups for a prime p, defined as inverse limits of finite p-groups.^[3] Equivalently, a profinite group is pro-p if every open normal subgroup has p-power index.^[17] The Sylow p-subgroups of a profinite group G are its maximal pro-p subgroups, and any two such subgroups are conjugate in G; moreover, every pro-p subgroup of G is contained in some Sylow p-subgroup.^[17] This extends Sylow's theorems from finite to profinite settings. Finiteness conditions in profinite groups often involve concepts like finite width, particularly for verbal subgroups. A non-trivial word w in the free group has finite width in a finitely generated pro-p group if the verbal subgroup w(G) is generated by finitely many values of w.^[18] This holds for every non-trivial word, implying that profinite groups of finite rank satisfy bounded width properties for such subgroups.^[18] In some cases, the entire group is generated by finitely many subgroups of finite index, reflecting a controlled algebraic structure.^[17] Every profinite group admits a descending chain of open normal subgroups G = N_0 \trianglerighteq N_1 \trianglerighteq \cdots such that each quotient N_i / N_{i+1} is finite, and the intersection \bigcap_{i=1}^\infty N_i = \{1\}.^[3] This chain can be refined to a composition series where the factors are finite simple groups, and such series are unique up to permutation and isomorphism of factors by the profinite analog of the Jordan-Hölder theorem.^[17] The quotients stabilize in the sense that the chief factors (minimal normal subgroups in the chain) determine the group's structure.

Special classes

Procyclic groups

A procyclic group is defined as a profinite group that is topologically generated by a single element, meaning there exists an element whose powers form a dense subgroup in the profinite topology.^[19] Equivalently, it is the inverse limit of a directed system of finite cyclic groups.^[20] The structure of procyclic groups is characterized by their isomorphism to closed subgroups of the profinite completion of the integers, denoted \hat{\mathbb{Z}} = \prod_p \mathbb{Z}_p, where the product runs over all primes p and \mathbb{Z}_p denotes the p-adic integers.^[19] In the pro-p case, they are isomorphic to closed subgroups of \mathbb{Z}_p. For example, the p-adic integers \mathbb{Z}_p themselves form a fundamental procyclic group.^[20] Finite procyclic groups are precisely the finite cyclic groups. Infinite procyclic groups decompose into their p-primary components, each of which is either finite cyclic or isomorphic to \mathbb{Z}_p.^[19] The Pontryagin dual of a procyclic group is a discrete torsion abelian group.^[19]

Projective profinite groups

In the category of profinite groups, a profinite group P is called projective if it satisfies the following lifting property: for every epimorphism \phi: G \twoheadrightarrow H of profinite groups and every continuous homomorphism f: P \to H, there exists a continuous homomorphism \tilde{f}: P \to G such that \phi \circ \tilde{f} = f. This definition mirrors the notion of projective objects in category theory, adapted to the setting of profinite groups with continuous homomorphisms. Free profinite groups are projective, and conversely, the projective profinite groups are precisely the free profinite groups generated by their (profinite) generating sets. Moreover, every profinite group arises as a continuous quotient of some projective profinite group; specifically, if G is generated by a (possibly infinite) set X, then G is a quotient of the free profinite group on X. From a cohomological perspective, a profinite group P is projective if and only if it has cohomological dimension at most 1, meaning that \operatorname{Ext}^1_{\mathcal{C}}(G, P) = 0 for every profinite group G, where \mathcal{C} denotes the category of profinite groups (or equivalently, the continuous cohomology group H^2(G, P) = 0 for all discrete profinite G-modules P). This cohomological vanishing captures the exactness of the Hom functor \operatorname{Hom}(-, P). A basic example of a projective profinite group is the free profinite group on a single generator, which is isomorphic to the profinite completion \hat{\mathbb{Z}} of the integers.

Free profinite groups

The free profinite group on a set S, denoted F_S, is defined as the profinite completion of the discrete free group on S.^[8] This construction equips F_S with the inverse limit topology, making it a compact, totally disconnected topological group that captures all finite quotient information of the underlying free group. The universal property of F_S states that for any profinite group G and any function f: S \to G, there exists a unique continuous homomorphism \phi: F_S \to G such that \phi(s) = f(s) for all s \in S.^[8] This property positions F_S as the free object in the category of profinite groups with respect to the set S, generalizing the universal mapping property of discrete free groups to the continuous setting. Structurally, F_S contains the discrete free group on S as a dense subgroup, with the only relations imposed arising from the finite quotients of this free group.^[8] Consequently, elements of F_S can be understood through their images in these finite quotients, ensuring that the topology reflects the residual finiteness of the free group. Free profinite groups are projective in the category of profinite groups.^[8] The rank of F_S, defined as the minimal cardinality of a topological generating set, equals |S|.^[8] For infinite S, F_S is uncountable, as its cardinality is at least $2^{|S|} due to the embedding into products of finite groups. Every profinite group admits a profinite presentation as a quotient F_S / N, where N is a closed normal subgroup of the free profinite group F_S on some set S.^[8] This representation underscores the generative role of free profinite groups in the category, analogous to presentations in discrete group theory but respecting the profinite topology.^[8]

Ind-finite groups

Ind-finite groups provide a conceptual counterpart to profinite groups within the category of topological groups. They are defined as the direct limits of directed inductive systems of finite groups, where the transition maps are injective homomorphisms. Formally, given a directed poset I and finite groups \{G_i\}_{i \in I} equipped with injective group homomorphisms \phi_{ij}: G_i \to G_j for i \leq j satisfying the compatibility conditions, the direct limit G = \varinjlim_{i \in I} G_i is the quotient of the disjoint union \coprod_{i \in I} G_i by the equivalence relation generated by g \sim \phi_{ij}(g) for g \in G_i and i \leq j. This construction yields G as an increasing union of the images of the finite groups G_i under the canonical maps to G.^[21]^[22] The natural topology on an ind-finite group is the inductive limit topology inherited from the discrete topologies on the component finite groups G_i. This results in an ind-discrete topology on G, which reduces to the discrete topology when the directed system is countable (e.g., indexed by the natural numbers). Unlike profinite groups, ind-finite groups are never compact unless they are finite, as the discrete topology on an infinite set precludes compactness. Moreover, infinite ind-finite groups lack proper open subgroups of finite index, reflecting their structure as unions of finite subgroups without global finite-index structure.^[21] In a categorical sense, the category of ind-finite groups is the ind-completion of the category of finite groups, standing in duality to the pro-completion that defines profinite groups. For abelian groups, this duality manifests concretely via Pontryagin duality, which pairs compact abelian profinite groups with their discrete torsion duals that are ind-finite. For instance, the Pontryagin dual of the additive group of p-adic integers \mathbb{Z}_p (a profinite group) is the Prüfer p-group \mathbb{Z}(p^\infty) = \varinjlim_n \mathbb{Z}/p^n\mathbb{Z}, a countable ind-finite torsion group.^[23]^[24] Every countable locally finite group embeds as a subgroup into an ind-finite group; notable examples include the finitary symmetric group \mathrm{FSym}(\mathbb{N}), the direct limit \varinjlim_n S_n of finite symmetric groups via inclusions fixing additional points, and the Prüfer p-groups. These groups highlight the non-compact, discrete nature of ind-finite structures, contrasting sharply with the compact topology and finite-index open subgroups characteristic of profinite groups.^[21]

Applications

In number theory

In class field theory, the idele class group C_K of a number field K is a key object that encodes arithmetic data through its quotient by the connected component of the identity, C_K^0, which yields a profinite group isomorphic to the inverse limit of the ray class groups of K.^[25] This profinite quotient C_K / C_K^0 is totally disconnected and compact, facilitating the study of abelian extensions via continuous homomorphisms.^[25] The Artin reciprocity map provides a canonical isomorphism from this profinite idele class group to the Galois group \mathrm{Gal}(K^{\mathrm{ab}}/K), establishing a precise correspondence between arithmetic ideals and Galois actions in the maximal abelian extension.^[25] Profinite groups also arise in the construction of p-adic L-functions and zeta functions, where they serve as domains for interpolation via continuous characters. For instance, the Kubota-Leopoldt p-adic zeta function \zeta_p(s) is defined as a p-adic measure on the profinite group \mathbb{Z}_p^\times, interpolating special values of the Riemann zeta function at negative integers twisted by Dirichlet characters.^[26] More generally, p-adic L-functions for Hecke characters of number fields are constructed using measures on profinite units groups, capturing analytic continuations and relations to arithmetic invariants like regulators and class numbers.^[26] In Iwasawa theory, profinite groups model infinite Galois extensions, particularly \mathbb{Z}_p-extensions, where the Galois group is isomorphic to \mathbb{Z}_p, a procyclic profinite group, acting on modules such as class groups or Selmer groups.^[27] These profinite Galois modules, often denoted X_\infty, are \Lambda-modules over the Iwasawa algebra \Lambda = \mathbb{Z}_p[[ \mathbb{Z}_p ]], allowing the study of growth rates and characteristic ideals that relate p-adic L-functions to arithmetic data via the main conjecture.^[27] A concrete example is the cyclotomic \mathbb{Z}_p-extension of \mathbb{Q}, denoted \mathbb{Q}_\infty, obtained as the fixed field of the torsion subgroup in \mathbb{Q}(\mu_{p^\infty})/\mathbb{Q}; its Galois group \mathrm{Gal}(\mathbb{Q}_\infty / \mathbb{Q}) \cong \mathbb{Z}_p is profinite, and the associated Iwasawa module for the class group vanishes, reflecting the triviality of class numbers in this tower.^[27]

In Galois theory

In infinite Galois theory, the Krull topology equips the Galois group of an algebraic extension with a structure that renders it profinite. Specifically, for a Galois extension L/K, the Krull topology on \Gal(L/K) is defined such that the open subgroups are precisely those corresponding to finite Galois subextensions of L/K, making \Gal(L/K) a compact, Hausdorff, and totally disconnected topological group, hence profinite.^[28] This topological framework, introduced by Krull in 1928, ensures that the classical Galois correspondence extends to infinite extensions by restricting to closed subgroups.^[29] The fixed fields correspondence in this setting relies on continuous actions of the profinite Galois group on the algebraic closure. For a profinite group G = \Gal(\overline{K}/K) acting continuously on \overline{K}, the map sending a closed subgroup H \leq G to its fixed field K^H = \{ x \in \overline{K} \mid \sigma(x) = x \ \forall \sigma \in H \} and conversely sending an intermediate field K \subseteq E \subseteq \overline{K} to \Gal(\overline{K}/E) establishes a bijection between closed subgroups of G and intermediate fields. This correspondence preserves the lattice structure, with normality and quotients corresponding appropriately, and open subgroups fixing finite extensions.^[4]^[10] A fundamental fact is that the Galois group of the separable closure \overline{K}^s/K is profinite, as it arises as the inverse limit of the finite Galois groups \Gal(E/K) over all finite separable extensions E/K. The finite quotients of this profinite group correspond exactly to the finite separable extensions, via the surjective maps \Gal(\overline{K}^s/K) \twoheadrightarrow \Gal(E/K) for finite E.^[4]^[10] Profinite cohomology provides tools to study these Galois groups through their actions on modules. For a profinite group G = \Gal(\overline{K}/K) and a discrete G-module M (endowed with the discrete topology and continuous action), the profinite cohomology groups H^i(G, M) are defined as the derived functors of the invariant functor on the category of discrete G-modules, computed via the continuous cochain complex or as direct limits over finite quotients: H^i(G, M) \cong \varinjlim H^i(G/U, M) where U runs over open normal subgroups. These groups capture obstructions to extensions and splitting problems in Galois theory.^[30] An illustrative example arises with decomposition groups in profinite Galois settings. In a Galois extension L/K with a discrete valuation, the decomposition group at a prime \mathfrak{p} of K above which \mathfrak{P} lies in L is the closed subgroup \mathcal{D}_\mathfrak{P} = \{ \sigma \in \Gal(L/K) \mid \sigma(\mathfrak{P}) = \mathfrak{P} \}, which is itself profinite as a closed subgroup of the profinite \Gal(L/K), and its quotient by the inertia subgroup yields the Galois group of the residue field extension.^[31]

In topology and algebra

Profinite groups play a significant role in topological group theory due to their inherent structure as compact, totally disconnected Hausdorff spaces. These spaces, often referred to as profinite spaces, are precisely the Stone spaces in the sense of Stone duality, which establishes a contravariant equivalence between the category of Boolean algebras and the category of compact totally disconnected Hausdorff spaces.^[32] Under this duality, profinite groups arise as topological groups on such spaces, and every profinite group is a continuous homomorphic image of a power set endowed with the product topology, reflecting the duality's connection to clopen partitions and ultrafilters. In abstract algebra, varieties of profinite groups are subclasses defined by group-theoretic laws (equations) that are preserved under the formation of profinite completions, meaning that if a discrete group satisfies a law in all its finite quotients, the law holds in the completion.^[33] These varieties are closed under taking closed subgroups, continuous quotients, and inverse limits, providing a framework analogous to varieties of finite groups but extended to the profinite setting. For instance, the variety of abelian profinite groups consists of those satisfying the commutator law [x,y]=1, which is detected uniformly across all finite quotients.^[32] The Burnside problem, concerning whether finitely generated groups of bounded exponent are finite, finds partial resolution through profinite methods, particularly in the restricted case where the exponent is fixed. Zelmanov's solution to the restricted Burnside problem relies on analyzing Lie rings associated to profinite completions, showing that such groups are finite.^[32] Moreover, in residually finite groups, finite quotients detect freeness: a finitely generated residually finite group is free if and only if its profinite completion is the free profinite group on the same number of generators, as the finite quotients faithfully capture the free relations.^[6] In topological dynamics, profinite groups admit continuous actions on Cantor sets, which are compact, totally disconnected metric spaces serving as universal models for such dynamics. These actions, often minimal and free, arise from embeddings into homeomorphism groups of the Cantor set and are studied for properties like topological entropy and orbit equivalence.^[34] A concrete example is the automorphism group of a profinite tree, which is the inverse limit of automorphism groups of finite trees and thus forms a profinite group acting naturally on the boundary Cantor set of the tree.^[35] Similarly, automorphism groups of profinite graphs, constructed as inverse limits over finite graph covers, yield profinite groups with rich dynamical properties on their associated spaces.^[35]

References

[1]
[PDF] INVERSE LIMITS AND PROFINITE GROUPS - OSU Math
Definition 3.1. A profinite group is a topological group which is obtained as the inverse limit of a collection of finite groups, each given the discrete ...Missing: authoritative sources
[2]
[PDF] Profinite Groups - of /websites
We define a profinite group to be a topological group which is isomorphic (as a topological group) to a projective limit of finite groups. One defines a ...Missing: authoritative | Show results with:authoritative
[3]
[PDF] Part III Profinite Groups - DPMMS
Feb 1, 2020 · Definition 1.2.7. A profinite group is the inverse limit of a system of finite groups. Our first task is to show that these limits actually ...
[4]
3.5 Profinite groups and infinite Galois theory - Kiran S. Kedlaya
Another way to construct profinite groups uses inverse limits (or projective limits or sometimes just limits ). 🔗. Definition 3.5.9. Suppose we are given a ...
[5]
[PDF] Profinite Groups
Definition 4.3: Let G be a profinite group and H a closed subgroup. Then the index p. 38. (G : H) of H in G is defined by (G : H)=l.c.m.{(G/U : HU/U)} = l.c ...
[6]
[PDF] Profinite and residual methods in geometric group theory Sam Hughes
Inverse limits and profinite spaces. Definition II.A.1.1 (Directed poset). A ... A pro-C group is a group which is the inverse limit of groups in C. A ...<|control11|><|separator|>
[7]
[PDF] 24 The idele group, profinite groups, infinite Galois theory
Dec 6, 2015 · Profinite groups that are isomorphic to their profinite completion as groups are said to be strongly complete; this is equivalent to requiring ...<|control11|><|separator|>
[8]
Profinite Groups - SpringerLink
The aim of this book is to serve both as an introduction to profinite groups and as a reference for specialists in some areas of the theory.
[9]
[PDF] 26 The idele group, profinite groups, infinite Galois theory
Dec 1, 2021 · Similarly, the absolute Galois group of Q has uncountably many subgroups of index 2 but Q has only countably many quadratic extensions, see ...
[10]
[PDF] Profinite Groups and Infinite Galois Theory - Abhijit Mudigonda
Profinite groups are also topological groups, and can be defined purely topologically. 1.1 Infinite Galois Theory. The study of profinite groups was ...
[11]
[PDF] 27 Local class field theory
Dec 6, 2021 · Theorem 27.2 implies that the Galois group of any finite abelian extension L/K of a local fields is canonically isomorphic to the quotient K ...Missing: Q_p^× | Show results with:Q_p^×
[12]
[PDF] Infinite Galois theory, Stone spaces, and profinite groups
E corresponds to the multiplicative group of p-adic integers congruent to 1 mod p; a subgroup of the full group of units. Example 3. Let p be a prime, and let k ...
[13]
[PDF] Compact seminar on Profinite Groups Introduction
Mar 15, 2017 · Theorem 2.13 G is compact Hausdorff and totally disconnected. Let U be an open normal subgroup of G. By Lemma 3.3 there is some Si = kerϕi ...Missing: compactness | Show results with:compactness
[14]
[PDF] lecture notes for course 88-909 cohomology of groups bar-ilan ...
(normal) if and only if it is an intersection of open (normal) subgroups. ... Let G be a profinite group and H G a closed normal subgroup. Consider the ...
[15]
[PDF] Profinite Groups
Definition (profinite group). A profinite group is the inverse limit of an inverse system of finite groups. Example. 1. The profinite completion of a ...Missing: constructive | Show results with:constructive
[16]
[PDF] 26 The idele group, profinite groups, infinite Galois theory
Dec 4, 2019 · profinite group whose open normal subgroups are precisely those of ... [E : K]=[G : N] and [E : F]=[H : N] (for the second equality ...
[17]
[PDF] Finiteness Properties of Profinite Groups - arXiv
A virtual automorphism of the profinite group G is a continuous isomorphism between open subgroups of G. ... In addition, cP(G) is finite if and only if cP(N) is ...
[18]
On the verbal width of finitely generated pro-p groups - Project Euclid
Abstract. Let p be a prime. It is proved that a non-trivial word w from a free group F has finite width in every finitely generated pro-p group.
[19]
Chernikov group - PlanetMath.org
Mar 22, 2013 · A Chernikov group is a group G that has a normal subgroup N such that G/N is finite and N is a direct product.Missing: profinite | Show results with:profinite
[20]
Chernikov group - Encyclopedia of Mathematics
Jan 15, 2017 · A group satisfying the minimum condition on subgroups and having a normal Abelian subgroup of finite index.Missing: definition | Show results with:definition
[21]
https://books.google.com/books/about/Locally_Finite_Groups.html?id=NPpwVBCxVgUC
[22]
[PDF] Profinite Groups, Exercise Sheet 1 Lent 2021 - DPMMS
13. Procyclic groups. A procyclic group is an inverse limit of finite cyclic groups. You have already seen the procyclic groups Z/mZ, Zp and bZ.
[23]
https://www-users.cse.umn.edu/~webb/RepBook/RepBookLatex.pdf
[24]
Locally Finite Groups - Google Books
Apr 1, 2000 · Contents ; CHAPTER 1 BASIC METHODS CONCEPTS AND EXAMPLES. 1 ; CHAPTER 2 CENTRALIZERS IN LOCALLY FINITE GROUPS. 68 ; CHAPTER 3 LOCALLY FINITE GROUPS ...
[25]
[PDF] Boolean and Profinite Loops - ScholarWorks at WMU
Ribes and Zalesski also prove the following result, which, when used in tandem with Proposition 108, describes a significant class of profinite groups. G for ...
[26]
[PDF] A Course in Finite Group Representation Theory
The representation theory of finite groups has a long history, going back to the 19th century and earlier. A milestone in the subject was the definition of ...
[27]
[PDF] Lectures on Algebraic Topology - MIT Mathematics
A graded abelian group is a sequence of abelian groups, indexed by the integers. ... It's called the Prüfer group (at p). Show that Zp∞ ∼. =Z[1/p]/Z and that. Q ...
[28]
[PDF] Class Field Theory
These notes contain an exposition of abelian class field theory using the algebraic/cohomological approach of Chevalley and Artin and Tate.
[29]
[PDF] AN INTRODUCTION TO p-ADIC L-FUNCTIONS
Let G be a profinite abelian group (e.g. G = Zp or G = Z× p , which are the examples of most interest to us). We denote by C (G, L) the space of continuous ...
[30]
[PDF] 1 Zp-extensions and ideal class groups.
The first isomorphism is just a consequence of the definition of the Galois group for an infinite Galois extension. The second isomorphism is defined by using ...<|control11|><|separator|>
[31]
[PDF] infinite galois theory (draft, ctnt 2020)
To fix the Galois correspondence for infinite-degree Galois extensions, Krull defined a topology on Galois groups so that there is a one-to-one correspondence ...
[32]
https://link.springer.com/book/10.1007/978-3-642-01642-4
[33]
[PDF] Math 210B. Profinite group cohomology 1. Motivation Let {Γ i} be an ...
Profinite groups are inverse limits of finite groups with surjective transition maps, and the related cohomology is a limit of ordinary group cohomologies.Missing: classified | Show results with:classified
[34]
[PDF] Chapter 7 Ramification theory
Since decomposition group, inertia group and ramification group are closed subgroups of GalL|K, Theorem 24.10 shows that they are equal to GalL|(L|K, v)d, GalL ...
[35]
Profinite Groups
### Summary of Profinite Groups by Ribes and Zalesski (Sections on Normal Subgroups, Open/Closed, Extensions, Cohomology, Divisible Subgroups)
[36]
Equations and algebraic geometry over profinite groups
Dec 3, 2010 · Neumann, Varieties of Groups, Springer, Berlin (1967). MATH Google Scholar. V. N. Remeslennikov, “Embedding theorems for profinite groups,” Izv.
[37]
[1809.08475] Galois groups and Cantor actions - arXiv
Sep 22, 2018 · In this paper, we study the actions of profinite groups on Cantor sets which arise from representations of Galois groups of certain fields of rational ...
[38]
Profinite Graphs and Groups | SpringerLink
This book offers a detailed introduction to graph theoretic methods in profinite groups and applications to abstract groups.