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Conversely, if G contains a proper non-trivial normal subgroup N, then the composition of the natural surjective group homomorphism G → G / N with the regular representation of G / N produces a representation π of G which has kernel N. Taking χ to be the character of some non-trivial subrepresentation of π, we have a character satisfying the hypothesis in the direct statement above.

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## Some Related Sentences

Conversely and if

__Conversely__

**,**

__if__statistics were uncovered

**which**contradicted

**a**cherished theory

**,**

**the**sources were denounced as faulty

**.**

__Conversely__

**,**experimenters would consider as impressive such statements as

**the**following

**,**

**which**

**,**

__if__they turned out

**to**

**be**hits

**,**are so unusual as

**to**

**be**really significant:

__Conversely__

**,**

__if__he gives

**a**heavy rating

**to**his own reading

**,**and finds more accurate facts

**in**it than

**in**

**the**others

**,**

**a**point is chalked up for

**the**intrinsic

**,**objective meaningfulness

**of**this type

**of**mediumistic material

**.**

__Conversely__

**,**

**a**subset R defines

**a**binary function

__if__and only

__if__

**,**for any x

**in**X and y

**in**Y

**,**there exists

**a**unique z

**in**Z such that ( x

**,**y

**,**z ) belongs

**to**R

**.**

__Conversely__

**,**

__if__

**a**Boolean ring A is given

**,**

**we**can turn it into

**a**Boolean algebra by defining x ∨ y := x + y + ( x · y ) and x ∧ y := x · y

**.**

__Conversely__

**,**

__if__

**the**second theory is correct

**,**

**then**

**the**Athenians were merely reacting

**to**

**the**Persians attacking them

**.**

__Conversely__

**the**equilibrium position is said

**to**

**be**" far

**to**

**the**left "

__if__hardly any product is formed from

**the**reactants

**.**

__Conversely__

**,**

__if__ƒ: C

**→**C is

**a**function

**which**is differentiable when regarded as

**a**function on R < sup > 2 </ sup >,

**then**ƒ is complex differentiable

__if__and only

__if__

**the**Cauchy – Riemann equations hold

**.**

__Conversely__

**,**

__if__

**the**amount

**of**fat is increased

**,**increasing

**the**number

**of**calories ingested will only add weight

**.**

__Conversely__

**,**

__if__

**the**source

**of**waves is moving away from

**the**observer

**,**each wave is emitted from

**a**position farther from

**the**observer than

**the**previous wave

**,**so

**the**arrival time between successive waves is increased

**,**reducing

**the**frequency

**.**

__Conversely__every partition

**of**comes from an equivalence relation

**in**this way

**,**according

**to**

**which**

__if__and only

__if__and belong

**to**

**the**same set

**of**

**the**partition

**.**

If neither A nor B includes

**the**idea**of**existence**,****then**"**some**A are B " simply adjoins A**to**B**.**__Conversely__**,**__if__A or B do include**the**idea**of**existence**in****the**way that " triangle "**contains****the**idea " three angles equal**to**two right angles ",**then**" A exists " is automatically true**,**and**we****have**an ontological proof**of**A's existence**.**__Conversely__

**,**

__if__

**the**measured potential is negative

**,**

**the**electrode is more oxidizing than

**the**SHE ( such as Zn

**in**ZnSO < sub > 4 </ sub > where

**the**standard electrode potential is − 0

**.**76 V ).

__Conversely__

**,**

__if__

**the**firm is able

**to**get bulk discounts

**of**an input

**,**

**then**it could

**have**economies

**of**scale

**in**

**some**range

**of**output levels even

__if__it

**has**decreasing returns

**in**production

**in**that output range

**.**

__Conversely__

**,**daylight stock shot

**in**daylight will look

**normal**

**,**but orange

__if__shot under tungsten lights

**.**

__Conversely__

**,**

__if__

**the**data are sparse — that is

**,**

__if__only K out

**of**

**N**Fourier coefficients are nonzero —

**then**

**the**complexity can

**be**reduced

**to**O ( K log

**N**log (

**N**

**/**K )), and this

**has**been demonstrated

**to**lead

**to**practical speedups compared

**to**an ordinary FFT for

**N**

**/**K > 32

**in**

**a**large-N example (

**N**= 2 < sup > 22 </ sup >) using

**a**probabilistic approximate algorithm (

**which**estimates

**the**largest K coefficients

**to**several decimal places ).

Conversely and G

__Conversely__

**,**given

**a**groupoid

__G__

**in**

**the**algebraic sense

**,**let

__G__< sub > 0 </ sub >

**be**

**the**set

**of**all elements

**of**

**the**form x * x < sup >− 1 </ sup >

**with**x varying through

__G__and define

__G__( x * x < sup >-1 </ sup >, y * y < sup >-1 </ sup >) as

**the**set

**of**all elements f such that y * y < sup >-1 </ sup > * f * x * x < sup >-1 </ sup > exists

**.**

__Conversely__

**,**

**if**F is left adjoint

**to**

__G__

**,**and

__G__is naturally isomorphic

**to**

__G__′

**then**F is also left adjoint

**to**

__G__′.

As stated earlier

**,**an adjunction between categories C and D gives rise**to****a**family**of**universal morphisms**,**one for each object**in**C and one for each object**in**D**.**__Conversely__**,****if**there exists**a**universal morphism**to****a**functor__G__: C**→**D from every object**of**D**,****then**__G__**has****a**left adjoint**.**__Conversely__

**,**every Polish space is homeomorphic

**to**

**a**

__G__< sub > δ </ sub >- subset

**of**

**the**Hilbert cube

**.**

__Conversely__

**,**one can show that given

**a**manifold M and

**a**principal H-bundle P over M

**,**and

**a**1-form η

**with**these properties

**,**

**then**P is locally isomorphic as an H-bundle

**to**

**the**principal homogeneous bundle

__G__

**→**

__G__

**/**H

**.**

The principal connection α on Q can

**be**recovered from**the**form η by taking Q**to****be****the**associated bundle P ×< sub > H </ sub >__G__**.**__Conversely__**,****the**form η can**be**recovered from α by pulling back along**the**inclusion P ⊂ Q**.**
Given

**then****a****normal**extension L**of**K**,****with**automorphism**group**Aut ( L**/**K ) =__G__**,**and containing α**,**any element g ( α ) for g**in**__G__will**be****a**conjugate**of**α**,**since**the**automorphism g sends roots**of**p**to**roots**of**p**.**__Conversely__any conjugate β**of**α is**of**this form:**in**other words**,**__G__acts transitively on**the**conjugates**.**
A section Γ

**of**dπ: TP**/**__G__**→**TM such that Γ: TM**→**TP**/**__G__is**a**linear morphism**of**vector bundles over M**,**can**be**identified**with****a**principal connection**in**P**.**__Conversely__**,****a**principal connection as defined**above**gives rise**to**such**a**section Γ**of**TP**/**__G__**.**__Conversely__

**,**given

**a**left kG module M

**,**

**then**M is

**a**k vector space

**,**and multiplication

**with**an element g

**of**

__G__yields

**a**k-linear automorphism

**of**M ( since g is invertible

**in**kG ),

**which**describes

**a**

**group**

**homomorphism**

__G__

**→**GL ( M ).

Conversely and contains

__Conversely__

**,**Scheme

__contains__multiple coherent subsets that suffice

**to**construct

**the**rest

**of**

**the**language as library macros

**,**and so

**the**language designers do not even bother

**to**say

**which**portions

**of**

**the**language must

**be**implemented as language constructs

**,**and

**which**must

**be**implemented as parts

**of**

**a**library

**.**

__Conversely__SVG is an ideal PDL for print-oriented uses

**,**as it

__contains__all

**the**functionality required

**to**place each glyph and image

**in**

**a**chosen location on

**the**final page

**.**

Each convex set containing X must ( by

**the**assumption that it is convex ) contain all convex combinations**of**points**in**X**,**so**the**set**of**all convex combinations is contained**in****the**intersection**of**all convex sets containing X**.**__Conversely__**,****the**set**of**all convex combinations is itself**a**convex set containing X**,**so it also__contains__**the**intersection**of**all convex sets containing X**,**and therefore**the**sets given by these two definitions must**be**equal**.**__Conversely__

**,**experiments

**with**robotic dummies were indeed able

**to**induce

**some**recruitment

**,**

**which**should not

**have**been possible

**if**

**the**dance

__contains__no information

**.**

__Conversely__

**,**

**the**Salidar

**group**

__contains__no members

**of**

**the**Red Ajah

**,**so

**the**Salidar Hall

**has**no Red sitters

**.**

Conversely and proper

__Conversely__

**,**

**the**vast majority

**of**early attempts at

__proper__excavation techniques failed

**to**accurately measure or record stratigraphy

**,**thereby failing

**to**provide

**a**secure context for artefact finds making interpretations extremely limited

**in**scope

**.**

__Conversely__

**,**

**the**digraph ⟨ gh ⟩ is never pronounced

**in**syllable onsets and is almost never pronounced

**in**syllable codas (

**the**

__proper__name Pittsburgh is an exception ).

__Conversely__

**,**

**the**severity

**of**these problems were minimized based on

__proper__and precise proximity

**of**

**the**wheels

**to**

**the**guitar strings

**.**

__Conversely__

**,**while Bayes rules

**with**respect

**to**

__proper__priors are virtually always admissible

**,**generalized Bayes rules corresponding

**to**improper priors need not yield admissible procedures

**.**

__Conversely__

**,**

**the**mechanical interaction between

**the**T-tubule's L-type calcium channel and

**the**calcium-release channel is critical

**to**

__proper__skeletal muscle contraction

**,**whereas it contributes little

**to**

**the**contraction

**of**cardiac muscle

**.**

I ’ m longing

**to**set me down shorely and raise my chaps__proper__( 50 – 51 ).”__Conversely__**,**her husband Brack is committed**to****the**mines and answers her as**the**family provider: “ It was never meant for**a**body**to****be**full content on**the**face**of**this earth**.**0.215 seconds.