In mathematics, a profinite integer is an element of the ring (sometimes pronounced as zee-hat or zed-hat) Z ^ = lim ← ⁡ Z / n Z , {\displaystyle {\widehat...

In mathematics, the term profinite is used for profinite groups, topological groups profinite sets, also known as "profinite spaces" or "Stone spaces"...

mathematics, a profinite group is a topological group that is in a certain sense assembled from a system of finite groups. The idea of using a profinite group...

related areas of mathematics, a Stone space, also known as a profinite space, profinite set, or Boolean space, is a compact Hausdorff totally disconnected...

In mathematics, a locally profinite group is a Hausdorff topological group in which every neighborhood of the identity element contains a compact open...

In mathematics, more precisely in formal language theory, the profinite words are a generalization of the notion of finite words into a complete topological...

with f? Analogously, an embedding problem for a profinite group F consists of the following data: Two profinite groups H and G and two continuous epimorphisms...

G-sets for a fixed profinite group G. For example, G might be the group denoted Z ^ {\displaystyle {\hat {\mathbb {Z} }}} (see profinite integer), which...

absolute Galois group is well-defined up to inner automorphism. It is a profinite group. When K {\displaystyle K} is a perfect field, K sep {\displaystyle...

In mathematics, a pro-p group (for some prime number p) is a profinite group G {\displaystyle G} such that for any open normal subgroup N ◃ G {\displaystyle...

A profinite group is called procyclic if it can be topologically generated by a single element. Examples of profinite groups include the profinite integers...

is residually finite. In particular, all profinite groups are residually finite. One example of a profinite group is the p-adic integers. If a group has...

abelian categories. A condensed set is a sheaf of sets on the site of profinite sets, with the Grothendieck topology given by finite, jointly surjective...

. A profinite identity is a pair u {\displaystyle u} and v {\displaystyle v} of profinite words. A semigroup S is said to satisfy the profinite identity...

infinite Galois group is the absolute Galois group, which is an infinite, profinite group defined as the inverse limit of all finite Galois extensions E /...

(1–3): 363–371. doi:10.1007/BF02808118. —— (2000). "Closed Subgroups of Profinite Groups". Proceedings of the London Mathematical Society. 81: 29–54. doi:10...

particular a “tiny” and a “small” universe, and look at sheaves on tiny profinite sets with values in small sets; they term these pyknotic sets. In our...

theory about finite permutation representations of groups G which are profinite groups. The gist of the theory is that the fiber functor Φ of the Galois...

the discrete topology). This makes G {\displaystyle G} a profinite group (in fact every profinite group can be realised as the Galois group of a Galois extension...

where Z ^ {\displaystyle {\hat {\mathbb {Z} }}} is the profinite integer ring with its profinite topology. It is homeomorphic to the rational numbers Q...

supervision of Angus Macintyre, with a dissertation on the model theory of profinite groups. She was Senior researcher and team director in Algebra and Geometry...

a group is residually X if it embeds into its pro-X completion (see profinite group, pro-p group), that is, the inverse limit of the inverse system...

Grothendieck conjectured that the algebraic fundamental group G of C, a profinite group, determines C itself (i.e., the isomorphism class of G determines...

finite index (the profinite topology); and there is another topology defined in the same way using only congruence subgroups. The profinite topology gives...

profinite set. This is used in Condensed mathematics to show that condensed sets may be described as functors from extremally disconnected profinite sets...

types Irrational numbers Fuzzy numbers Transcendental numbers p-adic numbers (p-adic solenoids) Profinite integers Normal numbers Classification List...

these spectra are studied. Such spaces are also useful in the study of profinite groups. The structure space of a commutative unital Banach algebra is...

1007/978-1-4020-5764-9_16. ISBN 978-1-4020-4843-2. Retrieved 2022-08-27. "Continuous and profinite combinatorics" (PDF). Archived (PDF) from the original on 2009-02-26....

Projective symplectic group Projective semilinear group Projective profinite group, a profinite group with the embedding property This disambiguation page lists...

group: a pro-l version, which is motivic a k-pro-unipotent version, and a profinite version, which is anabelian ,; These versions were jointly defined by...

groups of td-type, locally profinite groups, or t.d. groups). The compact case has been heavily studied – these are the profinite groups – but for a long...

theory is an algebraic group, whereas in algebraic Galois theory, it is a profinite group equipped with the Krull topology. In mathematics, the indefinite...

sequence Integer-valued function Mathematical symbols Parity (mathematics) Profinite integer More precisely, each system is embedded in the next, isomorphically...

finite groups exploits their connections with compact topological groups (profinite groups): for example, a single p-adic analytic group G has a family of...

disconnected Hausdorff space; such spaces are called Stone spaces (also profinite spaces). Conversely, given any Stone space X, the collection of subsets...

of p-adic numbers. The group Z p {\displaystyle \mathbb {Z} _{p}} is a profinite group; it is isomorphic to a subgroup of the product ∏ n ≥ 1 Z / p n {\displaystyle...

X^{sep}:=X\times _{k}k^{sep}} is connected) there is an exact sequence of profinite groups: 1 → π 1 ( X s e p , x ¯ ) → π 1 ( X , x ¯ ) → Gal ⁡ ( k s e p...

X correspond to continuous sets (or abelian groups) acted on by the (profinite) group G, and étale cohomology of the sheaf is the same as the group cohomology...

theorem. Mathematical Intelligencer 1992 (Online at Lenstra's Homepage). Profinite Fibonacci Numbers, December 2005, PDF Print Gallery (M. C. Escher) Prof...

In mathematics, the Iwasawa algebra Λ(G) of a profinite group G is a variation of the group ring of G with p-adic coefficients that take the topology...

group, though some aspects generalize to locally finite groups and to profinite groups. For a prime p, the p-core of a finite group is defined to be its...

discrete group G / G n {\displaystyle G/G_{n}} . A basic example is a profinite completion. The image of the canonical map G → G ^ {\displaystyle G\to...

homotopical object in simplicial profinite sets. This has a forgetful functor to the homotopy category of profinite simplicial sets. Artin, Michael; Mazur...

deformation ring is a ring that controls liftings of a representation of a profinite group (usually a Galois group) from a finite field to a local ring. In...

prime number). For example, the domain could be the p-adic integers Zp, a profinite p-group, or a p-adic family of Galois representations, and the image could...

{Z} _{p}} can be expanded out in terms of roots of unity instead of as profinite elements in ∏ F p {\displaystyle \prod \mathbb {F} _{p}} . We also set...

Zalesskii, Pavel Aleksandrovich (1990). "[Open subgroups of free profinite products over a profinite space of indices]". Doklady Akademii Nauk SSSR (in Russian)...

1966 from the University of Bonn for a dissertation on the theory of profinite groups. Wilfried Brauer and his wife Ute were two of the 19 founding members...

⁠ It is a subring of the ring of all algebraic integers. The ring of profinite integers ⁠ Z ^ , {\displaystyle {\widehat {\mathbb {Z} }},} ⁠ the (infinite)...

a\geq b} . These observations are pivotal for constructing the ring of profinite integers, which is given as an inverse limit of all such maps. Dedekind's...

1\ 0\ 1..._{!}} Combinatorial number system (also called combinadics) Profinite integers, which can be represented as infinite digit sequences in the...

( X , x ) {\displaystyle \pi _{1}^{\textrm {et}}(X,x)} is exactly the profinite completion of the usual topological fundamental group π 1 ( X an , x )...

Springer-Verlag. ISBN 0-387-90424-7. Zbl 0423.12016. Shatz, Stephen S. (1972). Profinite groups, arithmetic, and geometry. Annals of Mathematics Studies. Vol. 67...

term for "irreducible". smooth 1.  A smooth representation of a locally profinite group G is a complex representation such that, for each v in V, there...

number. Supernatural numbers are used to define orders and indices of profinite groups and subgroups, in which case many of the theorems from finite group...

was the Tate–Poitou duality result. Technically speaking, G may be a profinite group, in which case the definitions need to be adjusted to allow only...

the asymptotic distribution of prime numbers. profinite A profinite integer is an element in the profinite completion Z ^ {\displaystyle {\widehat {\mathbb...

is isomorphic to Z ^ {\displaystyle {\widehat {\mathbf {Z} }}} , the profinite completion of Z, the integers. Therefore, the perfect pairing (for any...

the harmonic analysis. The Cantor group is a protypical example of a profinite group. It is the inverse limit of the groups F 2 n {\displaystyle F_{2}^{n}}...

for the theory of branch groups. He has made important contributions to profinite group theory, and to the model theory of groups, and has established generation...

Association of America, ISBN 978-0-88385-511-9. Shatz, Stephen S. (1972), Profinite Groups, Arithmetic, and Geometry, Princeton University Press, ISBN 978-0-691-08017-8...

{\displaystyle A} be the set of normal subgroups of finite index. The profinite completion of E {\displaystyle E} is defined to be the inverse limit of...

{\displaystyle \mathbb {Z} } . The p-adic integers can also be extended to profinite integers Z ^ {\displaystyle {\widehat {\mathbb {Z} }}} , which can be...

algebraic geometry, model theory, the theory of finite groups and of profinite groups. Let K be a field and let G = Gal(K) be its absolute Galois group...

numbers The irrational numbers The p-adic numbers; more generally, all profinite groups are totally disconnected. The Cantor set and the Cantor space The...

category if it is equivalent to the category of finite G-sets for some profinite group G. 2.  For technical reasons, some authors (e.g., Stacks project...

elementary means, the group Gal(Fq) can be shown to be the Prüfer group, the profinite completion of Z. This statement subsumes the fact that the only algebraic...

subgroups of free products of profinite groups, Communications of Algebra 22 (1994), 1467-1494. M. Jarden, On free profinite groups of uncountable rank,...

{\displaystyle \operatorname {Aut} ({\overline {\mathbb {F} _{p}}},f)} is a profinite group called the Morava stabilizer group. The Lubin–Tate theory describes...

groups that includes the solenoids Pontryagin duality p-adic solenoid Profinite integer Hewitt, Edwin; Ross, Kenneth A. (1979). Abstract Harmonic Analysis...

absolute Galois group G {\displaystyle G} of a field K {\displaystyle K} is profinite, hence compact, and hence equipped with a normalized Haar measure. Let...

infinite degree over K; the Galois group G of A over K is an infinite profinite group, so a compact topological group, and it is abelian. The central...

the Galois group G {\displaystyle G} of L / K {\displaystyle L/K} is a profinite group equipped with the Krull topology. Since G {\displaystyle G} is compact...

small groups Modular representation theory Monstrous moonshine P-group Profinite group Representation theory of finite groups Aschbacher, Michael (2004)...

holomorphy of Artin L-functions. Because of the incompatibility of the profinite topology on GK and the usual (Euclidean) topology on complex vector spaces...

Krull–Akizuki theorem Krull–Schmidt theorem Krull topology, an example of the profinite group Krull's intersection, a theorem within algebraic ring theory that...

{\displaystyle n!} . More generally, for any topologically finitely generated profinite group G {\displaystyle G} there is an identity exp ⁡ ( ∑ H ⊂ G x [ G :...

fixed points of automorphisms of p-groups, and studied subgroup growth of profinite and discrete groups. From 1995 Shalev developed and applied probabilistic...

additive group Zp of p-adic integers, and constructions from it. In fact any profinite group is a compact group. This means that Galois groups are compact groups...

. A special case occurring in algebra and number theory is when G is profinite, for example the absolute Galois group of a field. The resulting cohomology...

Examples from other areas of algebra include the Frattini cover of a profinite group and the universal cover of a Lie group. If F is some family of modules...

information inherent in the classical fundamental group: the former is the profinite completion of the latter. The fundamental group of a root system is defined...

group of the maximal abelian extension of the rational numbers is a free profinite group of countable rank This disambiguation page lists mathematics articles...

primarily focusses on low-dimensional topology, hyperbolic manifolds and profinite groups. He proved that the figure-eight knot is the only knot whose complement...

group with high probability. They also explored related questions for profinite groups and settled several open problems. In 2016, Pyber and Endre Szabó...

_{q}}}/\mathbf {F} _{q}\right),} because this Galois group is isomorphic to the profinite integers Z ^ = lim ← n ⁡ Z / n Z , {\displaystyle {\widehat {\mathbf {Z}...

is called the absolute Galois group of K {\displaystyle K} . This is a profinite group and it is therefore endowed with its natural Krull topology. For...

G is often the absolute Galois group of a field, and in particular is profinite, and the open subgroups therefore correspond to the finite extensions...

Almeida, and M. V. Volkov. "Subword complexity of profinite words and subgroups of free profinite semigroups." International Journal of Algebra and Computation...

some ideal I {\displaystyle I} (Eisenbud, Theorem 7.7). Formal scheme Profinite integer Locally compact field Zariski ring Linear topology Quasi-unmixed...

E)} to vanish adds a key complexity to the theory. Suppose that G is a profinite group acting on a module A with a surjective homomorphism π from the G-module...

the size of any minimal generating set of H {\displaystyle H} . Those profinite groups with finite Prüfer rank are more amenable to analysis. Specifically...

Fesenko's noncommutative local class field theory for arithmetically profinite Galois extensions of local fields studies appropriate local reciprocity...

Let G be a finite group (in fact everything will work verbatim for a profinite group). Then for any two finite G-sets X and Y we can define an equivalence...

on modular forms and modular units, the idea of a "distribution" on a profinite group, and value distribution theory. He made a number of conjectures...

Procyclic may refer to: a term related to the profinite groups in mathematics In chemistry, a term referring to an acyclic compound that can be converted...

ordinary or inductive sets while "multiplying" them gives Stone spaces or profinite sets. The same choice exists for finite power set algebras as the duals...

Université Laval. Abercrombie, Alexander G. (1994). "Subgroups and subrings of profinite rings". Mathematical Proceedings of the Cambridge Philosophical Society...

=\operatorname {Gal} (k_{s}/k)=\varprojlim \operatorname {Gal} (k'/k)} , the profinite group of finite Galois extensions of k. Then A ↦ X A = { k -algebra homomorphisms ...

topology. The group Z ^ {\displaystyle {\widehat {\mathbb {Z} }}} is the profinite completion of integers with respect to its subgroups of finite index....

where Z ^ {\displaystyle {\hat {\mathbb {Z} }}} is the profinite integer ring with its profinite topology. The notion of an arithmetic progression makes...

variety of finite monoids if and only if it can be defined by a set of profinite identities. Pseudovarieties are of particular importance in the study...

finite semigroups. The variety of finite nilsemigroups is defined by the profinite equalities x ω y = x ω = y x ω {\displaystyle x^{\omega }y=x^{\omega }=yx^{\omega...

Springer-Verlag. ISBN 3-540-61990-9. Zbl 0902.12004. Shatz, Stephen S. (1972). Profinite groups, arithmetic, and geometry. Annals of Mathematics Studies. Vol. 67...

spectral sequences. The same statement holds if G {\displaystyle G} is a profinite group, N {\displaystyle N} is a closed normal subgroup and H ∗ {\displaystyle...

space E ∞ {\displaystyle {\mathcal {E}}^{\infty }} has the structure of a profinite-dimensional manifold. An elementary diffiety is a pair ( E ∞ , C ( E ∞...

automorphisms of Fib s ¯ {\displaystyle {\text{Fib}}_{\overline {s}}} form a profinite group, denoted π 1 ( S , s ¯ ) {\displaystyle \pi _{1}(S,{\overline {s}})}...

It may of course not itself be a Lie group: it may for example be a profinite group. Pontryagin duality Peter, F.; Weyl, H. (1927), "Die Vollständigkeit...

Jiming; Wang, Zixi (2022). "Distinguishing 4-dimensional geometries via profinite completions". Geometriae Dedicata. 216 (52) 52. arXiv:2011.03784. doi:10...

Fesenko, Ivan (1999), "On just infinite pro-p-groups and arithmetically profinite extensions", Journal für die reine und angewandte Mathematik, 517: 61–80...

F} are finite subsets of the set of all places. The profinite integers are defined as the profinite completion of the rings Z / n Z {\displaystyle \mathbb...

category of finite groups, then pro-C is equivalent to the category of profinite groups and continuous homomorphisms between them. The process of endowing...

Paris: Secrétariat mathématique, MR 0193122 Shatz, Stephen S. (1972). Profinite groups, arithmetic, and geometry. Annals of Mathematics Studies. Vol. 67...

one can take a projective limit of finite constant group schemes to get profinite group schemes, which appear in the study of fundamental groups and Galois...

classical topology. Since it is also totally disconnected, G(k) is a profinite group (but not finite). As a result, G(k) contains infinitely many normal...

doi:10.1090/S0002-9939-96-03311-4. Newelski, Ludomir (2002). "Small profinite structures". Transactions of the American Mathematical Society. 354 (3):...

Edited by Lars V. Ahlfors, Lipman Bers 1971-07-21 430 9780691080819 67 Profinite Groups, Arithmetic, and Geometry. Stephen S. Shatz 1972-03-21 264 9780691080178...

Lyndon–Hochschild–Serre spectral sequence H p(G/N, H q(N, A)) ⇒ H p+q(G, A) where G is a profinite group, N is a closed normal subgroup, and A is a discrete G-module. The...

open subgroup of the Weil group. (The resulting topology is "locally profinite".) For global fields of characteristic p > 0 {\displaystyle p>0} (function...

synonymous with G-fibration. prime decomposition profinite profinite homotopy theory; it studies profinite spaces. properly discontinuous Not particularly...

the ring of p-adic integers and the field of p-adic numbers. See also "profinite integer" for an example in the similar spirit. If R is not commutative...

the degree of the extension. Galois groups for infinite extensions are profinite groups. Kummer theory The Galois theory of taking nth roots, given enough...

{O}}}_{K}=\prod _{\mathfrak {p}}{\mathcal {O}}_{\mathfrak {p}}} be the profinite completion of O K {\displaystyle {\mathcal {O}}_{K}} , where p {\displaystyle...

scheme, it may be thought of as a union of schemes, and this is enough to profinitely apply geometric methods to study it. Let G be an algebraic group over...

ISBN 978-3-11-034199-7; Theorem 10.4.13 on p. 236 L. Ribes, and P. Zalesskii, Profinite groups. Second edition. Ergebnisse der Mathematik und ihrer Grenzgebiete...

Jean-Pierre Serre, Structure de certains pro-p-groupes, d'après Demuškin (profinite groups) Michael F. Atiyah, The index of elliptic operators on compact...

groupes compacts totalement discontinus, d'après Tate (group cohomology for profinite groups) Alexander Grothendieck, Technique de descente et théorèmes d'existence...