If E / F is a Galois extension, then Gal ( E / F ) can be given a topology, called the Krull topology, that makes it into a profinite group.

A non-compact generalization of a profinite group is a locally profinite group.

Formally, a profinite group is a Hausdorff, compact, and totally disconnected topological group: that is, a topological group that is also a Stone space.

Equivalently, one can define a profinite group to be a topological group that is isomorphic to the inverse limit of an inverse system of discrete finite groups.

* The group of p-adic integers Z < sub > p </ sup > under addition is profinite ( in fact procyclic ).

The topology on this profinite group is the same as the topology arising from the p-adic valuation on Z < sub > p </ sup >.

Waterhouse showed that every profinite group is isomorphic to one arising from the Galois theory of some field K ; but one cannot ( yet ) control which field K will be in this case.

* The automorphism group of a locally finite rooted tree is profinite.

* Every closed subgroup of a profinite group is itself profinite ; the topology arising from the profiniteness agrees with the subspace topology.

If N is a closed normal subgroup of a profinite group G, then the factor group G / N is profinite ; the topology arising from the profiniteness agrees with the quotient topology.

* Since every profinite group G is compact Hausdorff, we have a Haar measure on G, which allows us to measure the " size " of subsets of G, compute certain probabilities, and integrate functions on G.

* A subgroup of a profinite group is open if and only if it is closed and has finite index.

* According to a theorem of Nikolay Nikolov and Dan Segal, in any topologically finitely-generated profinite group ( that is, a profinite group that has a dense finitely-generated subgroup ) the subgroups of finite index are open.

There is a notion of ind-finite group, which is the concept dual to profinite groups ; i. e. a group G is ind-finite if it is the direct limit of an inductive system of finite groups.

* If X is the spectrum of a field K with absolute Galois group G, then étale sheaves over 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 of G, i. e. the Galois cohomology of K.

Such spaces are also useful in the study of profinite groups.

In mathematics, profinite groups are topological groups that are in a certain sense assembled from finite groups ; they share many properties with their finite quotients.

* Finite groups are profinite, if given the discrete topology.

* The Galois theory of field extensions of infinite degree gives rise naturally to Galois groups that are profinite.

* The fundamental groups considered in algebraic geometry are also profinite groups, roughly speaking because the algebra can only ' see ' finite coverings of an algebraic variety.

The fundamental groups of algebraic topology, however, are in general not profinite.

* Every product of ( arbitrarily many ) profinite groups is profinite ; the topology arising from the profiniteness agrees with the product topology.

The inverse limit of an inverse system of profinite groups with continuous transition maps is profinite and the inverse limit functor is exact on the category of profinite groups.

Further, being profinite is an extension property.

The homomorphism η is characterized by the following universal property: given any profinite group H and any group homomorphism f: G → H, there exists a unique continuous group homomorphism g: G < sup >^</ sup > → H with f

The fundamental result of class field theory states that the group G is naturally isomorphic to the profinite completion of the idele class group C < sub > K </ sub > of K with respect to the natural topology on C < sub > K </ sub > related to the specific structure of the field K. Equivalently, for any finite Galois extension L of K, there is an isomorphism

Galois groups for infinite extensions are profinite groups.

In the above example, a connection with classical Galois theory can be seen by regarding as the profinite Galois group Gal (< span style =" text-decoration: overline "> F </ span >/ F ) of the algebraic closure < span style =" text-decoration: overline "> F </ span > of any finite field F, over F. That is, the automorphisms of < span style =" text-decoration: overline "> F </ span > fixing F are described by the inverse limit, as we take larger and larger finite splitting fields over F. The connection with geometry can be seen when we look at covering spaces of the unit disk in the complex plane with the origin removed: the finite covering realised by the z < sup > n </ sup > map of the disk, thought of by means of a complex number variable z, corresponds to the subgroup n. Z of the fundamental group of the punctured disk.

However, 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 representations or in the theory of the fundamental group scheme, and these are affine of infinite type.

In this case, the Galois group G of L / K is a profinite group equipped with the Krull topology.

The absolute Galois group G ( F < sub > q </ sub >) of a finite field, for example, is isomorphic to, the profinite completion of Z, the integers.

For any Boolean algebra B, S ( B ) is a compact totally disconnected Hausdorff space ; such spaces are called Stone spaces ( also profinite spaces ).

In fact any profinite group is a compact group.

Note that, as profinite groups are compact, the open subgroups are exactly the subgroups of finite index, so that the discrete quotient group is always finite.

Compound comparisons typically compare two sets of groups means where one set has two or more groups ( e. g., compare average group means of group A, B and C with group D ).

This group also includes some alkaloids that besides nitrogen heterocycle contain terpene ( e. g., evonine ) or peptide fragments ( e. g. ergotamine ).

People denotes a group of humans, either with unspecified traits, or specific characteristics ( e. g. the people of Spain or the people of the Plains ).

Monosaccharides can be grouped into aldoses ( having an aldehyde group at the end of the chain, e. g. glucose ) and ketoses ( having a keto group in their chain ; e. g. fructose ).

Whenever the book was written and whatever the historicity of the events recounted in it, clearly by the time it was written the term " Yehudim " ( יהודים-Jews ) already gained a meaning quite close to what it means up to the present — i. e. an ethnic-religious group, scattered in many countries, organised in autonomous communities and a target of hatred.

Male zebrafish are furthermore known to respond to more pronounced markings on females, i. e., " good stripes ", but in a group, males will mate with whichever females they can find.

There have been attempts at categorizing this fictional group of beings, and Phillip A. Schreffler argues that by carefully scrutinizing Lovecraft's writings a workable framework emerges that outlines the entire " pantheon " – from the unreachable " Outer Ones " ( e. g. Azathoth, who apparently occupies the centre of the universe ) and " Great Old Ones " ( e. g. Cthulhu, imprisoned on Earth in the sunken city of R ' lyeh ) to the lesser castes ( the lowly slave shoggoths and the Mi-go ).

On the map of Ptolemy, the " Kimbroi " are placed on the northernmost part of the peninsula of Jutland., i. e. in the modern landscape of Himmerland south of Limfjorden ( since Vendsyssel-Thy north of the fjord was at that time a group of islands ).

Besides immune competent cells ( granulocyte, monocyte, lymphocyte ) a large group of cells-considered previously to be fixed into tissues-are also motile in special physiological ( e. g. mast cell, fibroblast, endothelial cells ) or pathological conditions ( e. g. metastases ).

* Anti-Somozistas who had supported the revolution but felt betrayed by the Sandinista government – e. g. Edgar Chamorro, prominent member of the political directorate of the FDN, or Jose Francisco Cardenal, who had briefly served in the Council of State before leaving Nicaragua out of disagreement with the Sandinista government's policies and founding the Nicaraguan Democratic Union ( UDN ), an opposition group of Nicaraguan exiles in Miami.

For example in a paper reporting on a study involving human subjects, there typically appears a table giving the overall sample size, sample sizes in important subgroups ( e. g., for each treatment or exposure group ), and demographic or clinical characteristics such as the average age, the proportion of subjects of each sex, and the proportion of subjects with related comorbidities.

:: Sometime the term multi-database is used as a synonym to federated database, though it may refer to a less integrated ( e. g., without an FDBMS and a managed integrated schema ) group of databases that cooperate in a single application.

* Maximal possible number of writes ( of any specific bit or specific group of bits ; could be constrained by the technology used ( e. g., " write once " or " write twice "), or due to " physical bit fatigue ," loss of ability to distinguish between the 0, 1 states due to many state changes ( e. g., in Flash memory )).

Thus the probability of two failures in a same RAID group in time proximity is much smaller ( approximately the probability squared, i. e., multiplied by itself ).