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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.

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## Some Related Sentences

Every and subgroup

*

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

__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__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__

**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 free

__Every__

__free__man

**,**born in lawful wedlock

**,**and neither excommunicate nor outlaw

**,**was eligible

**for**membership

**.**

__Every__other province had

**the**freedom to regulate

**the**religious question

**as**it wished

**,**although

**the**Union stated every person should be

__free__in

**the**choice

**of**his personal religion and no person should be prosecuted based on his or her religious choice

**.**

The Body

**of**Liberties adopted in 1641 by**the**Massachusetts Bay colonists states**,**“__Every__married woman shall be__free__from bodily correction or stripes by her husband**,**unless it be in his own defense from her assault .” In**the**United States**,**legal decisions in Mississippi ( 1824 ) and North Carolina ( 1868 and 1874 ) make reference to — and reject — an unnamed " old doctrine " or " ancient law " by**which****a**man was allowed to beat his wife with**a**stick no wider than his thumb**.**__Every__antiseptic

**,**however good

**,**

**is**more or less toxic and irritating to

**a**wounded surface ;

**as**

**a**result

**,**in surgery

**,**

**the**antiseptic method has been replaced by aseptic method

**,**

**which**

**is**preventative in nature and relies on keeping

__free__from

**the**invasion

**of**bacteria rather than destroying them when present

**.**

* The Empty Space:

__Every__Friday**the**Students ' Union hosts**a**live music venue in The Empty Space**,**attended by students**for**__free__( this was canceled in 2011 due to changes installed by**the**69th Student Legislative Council ).
*

__Every__August**the**residents**of**Sleepy Eye host**the**annual " Corn Days " event**,**where__free__buttered corn**is**provided**,****as**well**as**live music**,****a**flea market**,**parade**,**and various other events**.**
*

__Every____free__**abelian****group****is**torsion-free**,**but**the**converse**is**not true**,****as****is**shown by**the**additive**group****of****the**rational numbers Q**.**__Every__week

**of**

**the**Michaelmas and Lent terms

**,**and twice in Easter term

**,**Churchill

**is**host to Pav

**,**

**a**music event unusual

**for**Cambridge events in that it

**is**

__free__and open to all university members

**.**

__Every__summer

**,**

**a**

__free__tourist train in

**the**city centre connects

**the**ancient parts

**of**

**the**city with

**the**government district

**.**

__Every__town wanted to be

**a**railway centre

**,**and

**the**government had great confidence in

**the**ability

**of**

**the**

__free__market to provide low freight rates to

**the**province's farmers provided sufficient charters were issued to competing companies

**.**

__Every__month Hustler

**is**mailed

**,**uninvited and

**for**

__free__

**,**to some

**of**

**the**offices

**of**Members

**of**

**the**United States Congress

**.**

__Every__Riemann surface

**is**

**the**quotient

**of**

**a**

__free__

**,**proper and holomorphic action

**of**

**a**discrete

**group**on its universal covering and this universal covering

**is**holomorphically isomorphic ( one also says: " conformally equivalent ") to one

**of**

**the**following:

__Every__vector space

**is**

__free__

**,**and

**the**

__free__vector space on

**a**set

**is**

**a**special case

**of**

**a**

__free__module on

**a**set

**.**

__Every__Ontario resident with his or her primary and permanent home in Ontario

**is**entitled to access emergency and preventive medical care ( although Bariatric surgery in many cases

**is**not covered ) under OHIP

__free__

**of**charge

**.**

Every and abelian

__Every__algebraic curve C

**of**genus g ≥ 1

**is**associated with an

__abelian__variety J

**of**dimension g

**,**by means

**of**an analytic map

**of**C into J

**.**

A finitely-generated

__abelian__**group****is**indecomposable if and only if it**is**isomorphic to Z or to**a**factor**group****of****the**form**for**some prime number p and some positive integer n**.**__Every__finitely-generated__abelian__**group****is****a**direct sum**of**( finitely many ) indecomposable__abelian__**groups****.**__Every__

__abelian__category

**is**an exact category if we just use

**the**standard interpretation

**of**" exact ".

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