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 is...
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adeles. In addition, it provides a basic tractable example of a profinite group. The profinite integers Z ^ {\displaystyle {\widehat {\mathbb {Z} }}} can be...
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In mathematics, a locally profinite group is a Hausdorff topological group in which every neighborhood of the identity element contains a compact open...
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it into a profinite group. Fundamental theorem of Galois theory Absolute Galois group Galois representation Demushkin group Solvable group Some authors...
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In mathematics, the term profinite is used for profinite groups, topological groups profinite sets, also known as "profinite spaces" or "Stone spaces"...
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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...
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closure of K that fix K. The absolute Galois group is well-defined up to inner automorphism. It is a profinite group. (When K is a perfect field, Ksep is the...
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hyperbolic group is virtually cyclic. A profinite group is called procyclic if it can be topologically generated by a single element. Examples of profinite groups...
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Stone space (redirect from Profinite space)
related areas of mathematics, a Stone space, also known as a profinite space or profinite set, is a compact Hausdorff totally disconnected space. Stone...
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finite groups, called a profinite group. For example, the group Z {\displaystyle \mathbb {Z} } p of p-adic integers and the absolute Galois group of a field...
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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...
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unitary group Projective symplectic group Projective semilinear group Projective profinite group, a profinite group with the embedding property This disambiguation...
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of finite groups exploits their connections with compact topological groups (profinite groups): for example, a single p-adic analytic group G has a family...
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finite groups Modular representation theory Monstrous moonshine Profinite group Finite ring Commuting probability Finite State Machine Infinite group Aschbacher...
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{\displaystyle X} . The algebraic fundamental group, as it is typically called in this case, is the profinite completion of π 1 ( X ) {\displaystyle \pi...
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Krull–Akizuki theorem Krull–Schmidt theorem Krull topology, an example of the profinite group Krull's intersection, a theorem within algebraic ring theory that describes...
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. Grothendieck conjectured that the algebraic fundamental group G of C, a profinite group, determines C itself (i.e., the isomorphism class of G determines...
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constructions from it. In fact any profinite group is a compact group. This means that Galois groups are compact groups, a basic fact for the theory of algebraic...
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Functor in category theory Profinite group – Topological group that is in a certain sense assembled from a system of finite groups Ultrafilter lemma – Maximal...
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k-simple group. As mentioned above, G(k) is compact in the classical topology. Since it is also totally disconnected, G(k) is a profinite group (but not...
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topology is called the profinite topology on G. A group is residually finite if, and only if, its profinite topology is Hausdorff. A group whose cyclic subgroups...
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is compact, again a consequence of Tychonoff's theorem. A profinite group (e.g. Galois group) is compact. Compactly generated space Compactness theorem...
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Fundamental theorem of Galois theory (category Theorems in group theory)
{\displaystyle G} a profinite group (in fact every profinite group can be realised as the Galois group of a Galois extension, see for example ). Note that...
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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...
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Embedding problem (category Group theory)
Analogously, an embedding problem for a profinite group F consists of the following data: Two profinite groups H and G and two continuous epimorphisms...
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Tannakian formalism (category Algebraic groups)
K. The group of natural transformations of Φ to itself, which turns out to be a profinite group in the Galois theory, is replaced by the group G of natural...
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=\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 ...
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projective limit of finite constant group schemes to get profinite group schemes, which appear in the study of fundamental groups and Galois representations or...
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modular forms and modular units, the idea of a "distribution" on a profinite group, and value distribution theory. He made a number of conjectures in...
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arbitrary groups. In this section G will denote a finite group, though some aspects generalize to locally finite groups and to profinite groups. For a prime...
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