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...

adeles. In addition, it provides a basic tractable example of a profinite group. The profinite integers Z ^ {\displaystyle {\widehat {\mathbb {Z} }}} can be...

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

it into a profinite group. Fundamental theorem of Galois theory Absolute Galois group Galois representation Demushkin group Solvable group Some authors...

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...

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...

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...

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...

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

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...

finite groups Modular representation theory Monstrous moonshine Profinite group Finite ring Commuting probability Finite State Machine Infinite group Aschbacher...

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

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...

{\displaystyle X} . The algebraic fundamental group, as it is typically called in this case, is the profinite completion of π 1 ( X ) {\displaystyle \pi...

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

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

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...

is compact, again a consequence of Tychonoff's theorem. A profinite group (e.g. Galois group) is compact. Compactly generated space Compactness theorem...

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...

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...

{\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...

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...

Functor in category theory Profinite group – Topological group that is in a certain sense assembled from a system of finite groups Ultrafilter lemma – Maximal...

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 consisting...

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...

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...

projective limit of finite constant group schemes to get profinite group schemes, which appear in the study of fundamental groups and Galois representations or...

=\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 ...

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...