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

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## Some Related Sentences

Every and closed

__Every__field has an algebraic extension which

**is**algebraically

__closed__( called its algebraic closure ), but proving this in general requires some form

**of**

**the**axiom

**of**choice

**.**

*****

__Every__unital real Banach algebra

**with**no zero divisors, and in which every principal ideal

**is**

__closed__,

**is**isomorphic to

**the**reals,

**the**complexes, or

**the**quaternions

**.**

__Every__character

**is**automatically continuous

**from**A to C, since

**the**kernel

**of**

**a**character

**is**

**a**maximal ideal, which

**is**

__closed__

**.**

*****

__Every__continuous map

**from**

**a**compact space to

**a**Hausdorff space

**is**

__closed__and proper ( i

**.**e.,

**the**pre-image

**of**

**a**compact set

**is**compact

**.**

__Every__open

**subgroup**H

**is**also

__closed__, since

**the**complement

**of**H

**is**

**the**open set given by

**the**union

**of**open sets gH for g in G

__Every__map that

**is**injective, continuous and either open or

__closed__

**is**an embedding

**;**however there are also embeddings which are neither open nor

__closed__

**.**

__Every__base he

__closed__resulted in

**a**new construction project elsewhere to expand another base, relocation

**of**forces projects and other related spending

**.**

__Every__year

**the**central business district (

**with**corners at

**the**Municipal Building, Grand Street Fire House and Croton-Harmon High School )

**is**

__closed__to automobile traffic for music, American food, local fund raisers, traveling, and local artists

**.**

The generalized Poincaré conjecture states that

__Every__simply connected,__closed__n-manifold**is**homeomorphic to**the**n-sphere**.**__Every__

__closed__3-manifold has

**a**prime decomposition: this means it

**is**

**the**connected sum

**of**prime three-manifolds ( this decomposition

**is**essentially unique except for

**a**small problem in

**the**case

**of**non-orientable manifolds ).

:

__Every__oriented prime__closed__3-manifold can be cut along tori, so that**the**interior**of**each**of****the**resulting manifolds has**a**geometric structure**with**finite volume**.*******

__Every__integrable subbundle

**of**

**the**tangent bundle — that

**is**, one whose sections are

__closed__under

**the**Lie bracket — also defines

**a**Lie algebroid

**.**

*****

__Every__irreducible

__closed__subset

**of**P < sup > n </ sup >( k )

**of**codimension one

**is**

**a**hypersurface

**;**i

**.**e.,

**the**zero set

**of**some homogeneous polynomial

**.**

__Every__October

**the**high street

**is**

__closed__for

**the**two Saturdays either side

**of**11 October for

**the**Marlborough Mop Fair

**.**

__Every__

__closed__point

**of**Hilb ( X ) corresponds to

**a**

__closed__subscheme

**of**

**a**fixed scheme X, and every

__closed__subscheme

**is**represented by such

**a**point

**.**

__Every__

__closed__curve c on X

**is**homologous to for some simple

__closed__curves c < sub > i </ sub >, that

**is**,

Every and subgroup

*****

__Every__finitely generated

**group**

**with**

**a**recursively enumerable presentation and insoluble word problem

**is**

**a**

__subgroup__

**of**

**a**finitely presented

**group**

**with**insoluble word problem

__Every__

__subgroup__

**of**

**a**topological

**group**

**is**

**itself**

**a**topological

**group**when given

**the**

**subspace**

**topology**

**.**

__Every__

__subgroup__

**of**

**a**free abelian

**group**

**is**

**itself**free abelian, which

**is**important for

**the**description

**of**

**a**general abelian

**group**as

**a**cokernel

**of**

**a**homomorphism between free abelian groups

**.**

__Every__

**group**

**of**prime order

**is**cyclic, since Lagrange's theorem implies that

**the**cyclic

__subgroup__generated by

Iwasawa worked

**with**so-called-extensions: infinite extensions**of****a**number field**with**Galois**group**isomorphic to**the**additive**group****of**p-adic integers for some prime p**.**__Every__**closed**__subgroup__**of****is****of****the**form, so by Galois theory, a-extension**is****the**same thing as**a**tower**of**fields such that**.*******

__Every__triangle

**group**T

**is**

**a**discrete

__subgroup__

**of**

**the**isometry

**group**

**of**

**the**sphere ( when T

**is**finite ),

**the**Euclidean plane ( when T has

**a**Z + Z

__subgroup__

**of**finite index ), or

**the**hyperbolic plane

**.**

Recall that

**a**subsemigroup G**of****a**semigroup S**is****a**__subgroup__**of**S ( also called sometimes**a****group**in S ) if there exists an idempotent e such that G**is****a****group****with**identity element e**.**A semigroup S**is**group-bound if some power**of**each element**of**S lies in some__subgroup__**of**S**.**__Every__finite semigroup**is**group-bound, but**a**group-bound semigroup might be infinite**.**__Every__

__subgroup__

**is**organized around

**a**set

**of**brothers, each

**of**whom

**is**often married to

**a**

**group**

**of**sisters

**.**

__Every__finite

**group**

**is**

**a**

__subgroup__

**of**

**the**mapping class

**group**

**of**

**a**

**closed**, orientable surface, moreover one can realize any finite

**group**as

**the**

**group**

**of**isometries

**of**some compact Riemann surface

**.**

__Every__proper

__subgroup__

**of**G can be assumed

**a**solvable

**group**, meaning that much theory

**of**such subgroups could be applied

**.**

__Every__quasinormal

__subgroup__

**is**

**a**modular

__subgroup__, that

**is**,

**a**modular element in

**the**lattice

**of**subgroups

**.**

__Every__

**group**has

**itself**(

**the**improper

__subgroup__) and

**the**trivial

__subgroup__as two

**of**its fully characteristic subgroups

**.**

Every and profinite

*****

__Every__product

**of**( arbitrarily many )

__profinite__groups

**is**

__profinite__

**;**

**the**

**topology**

**arising**

**from**

**the**

**profiniteness**

**agrees**

**with**

**the**product

**topology**

**.**

Every and group

Group actions / representations:

__Every____group__G can be considered as**a**category**with****a**single object whose morphisms are**the**elements**of**G**.**A functor**from**G to Set**is**then nothing but**a**__group__action**of**G on**a**particular set, i**.**e**.****a**G-set**.**__Every__galaxy

**of**sufficient mass in

**the**Local Group has an associated

__group__

**of**globular clusters, and almost every large galaxy surveyed has been found to possess

**a**system

**of**globular clusters

**.**

*****

__Every__Lie

__group__

**is**parallelizable, and hence an orientable manifold ( there

**is**

**a**bundle isomorphism between its tangent bundle and

**the**product

**of**

**itself**

**with**

**the**tangent space at

**the**identity )

__Every__instrumental

__group__( or section ) has

**a**principal who

**is**generally responsible for leading

**the**

__group__and playing orchestral solos

**.**

While he admits

**the**existence**of**caste-based discrimination, he writes that "__Every__social__group__cannot be regarded as**a**race simply because we want to protect it against prejudice and discrimination ".__Every__summer

**the**

__group__gathers in Newport, RI for week long dance training, seaside teas, and evenings enjoying

**the**splendors

**of**

**the**Gilded Age

**.**

__Every__synset contains

**a**

__group__

**of**synonymous words or collocations (

**a**collocation

**is**

**a**sequence

**of**words that go together to form

**a**specific meaning, such as " car pool "); different senses

**of**

**a**word are in different synsets

**.**

__Every__

__group__can be trivially made into

**a**topological

__group__by considering it

**with**

**the**discrete

**topology**

**;**such groups are called discrete groups

**.**

__Every__topological

__group__can be viewed as

**a**uniform space in two ways

**;**

**the**left uniformity turns all left multiplications into uniformly continuous maps while

**the**right uniformity turns all right multiplications into uniformly continuous maps

**.**

__Every__ten years, when

**the**general census

**of**population takes place, each citizen has to declare which linguistic

__group__they belong or want to be aggregated to

**.**

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