characterization of a class of torsion free groups in terms of endomorphisms

8/3/2019 Characterization of a Class of Torsion Free Groups in Terms of Endomorphisms

1/16

[ I O D J D O ANW\FIE NJ CIR@GC IRDO_

Zne. =4, Fn .


2/16

?2< G. J .

O N \ F G E D W _ ,

A\ .

[rnnj.

D tsujjdogs tn onfsdbgr / g

G(M)

wdt` ; 5

r ( /

f

M )

3r(j

f+9

M) 3

. . 5 nn jnr sncg pnsdtdvg dftgmgr f. j

f

1 j

f

M j

f

M ds t`us i

quisd iutncnrp`dsc nj j

f

M X4, p. 77?, R`. 2.


3/16

O@ I \I ORG\D QI RD N F NJ IOEI__NJRN\_DNF J \ G G M \ NW [ _ ?2?

M *(r )3I*( )+L * )

4

jnr

gvgry typg

.

(< )

_uppnsg

R(M)ds

edfgirey

nrbgrgb.

D j i ifb l irggegcgfts

nj M

suo`

t`it t{i)5t(al), t gf t(i+l)3t(i).

(?)

Dj

R{M)

ds

edfgirey nrbgrgb,

t`gf

jnr

gvgry typg

,

M*( )

ds

i purg

sulmrnup

nj M ifb@*( )3 @ M*( )jnr @purg df M.

[rnnj. (9)`nebs lgoiusg typg ds i quisd dsncnrp`dsc dfvirdif t

ifb(I+L)( )3 I( )+7(r),(I +7)*( )3I*( )+7*(r) X9,p. =7, ?S.

Df (


4/16

?22 G . J .O N \ F G E D W _ ,A\ .

[rnnj. _uppnsg

t`g

fnfzgrn gegcgfts

iifb njM

`ivg bdstdfot

typgs.

Dj

t{i

) 3

( ), t`gf

( )3

((

) + ) V ( )

t( )

3

t( )

.

Dj

(

) ifb (&) irg

bdstdfot, sncg

/ g

G(M) cips

tn ; ifb

tnfl,f 6;. R`gf

t(i)V

(/ )

3t(jl)3

(w )

3

( )

.

J n r

igM, i

bgfntgs

t`g

onsgt

nj i df

sncg spgodjdgb quntdgft

mrnup.

EGCCI


5/16

O @ I \ I O R G \ D Q I R D N F

NJ I OEI__ NJ RN\_DNF

J \ G G

M \ N W [ _

?27

d 3 9, ,a , jnr i sdfmeg pnsdtdvg dftgmgr f. Jnr ly bgjdfdtdnf, t`grg

gxdsts

jd :\ suo` t` it /,(&5 3 c , c

t

6 ;, ifb / V 3 ; jnr V d,

d 3 9, ,a . Fnw / 3 d d DE*d V / * g

sitdsjdgs

ji

t

3 (c

x

rfV&d,

d

3 9, , d .

(


6/16

?2: G . J .ON\FGEDW_,A\ .

p`dscs nj M ifb egt i

dj

,i

h

lgot dfbgpgfbgft gegcgfts nj M

wdt

(

x

)

5A 5At i

h

) . R`gf

t`grg gxdsts

j g \ suo t`it ji

fi

ej

f i

pnsdtdvg dftgmgr, ifb ji

d

3;,d 3


7/16

O@I\IORG\DQIRDNF NJ I OEI__NJ RN\_DNF J\GG M\NW[_ ?2=

t`i t

t`g

rgstrdotdnf

nj / tn I ds if

gfbncnrp`dsc

nj I ifb

t`us

j/c

ds i quisd-gfbncnrp`dsc cihdfm

L

i quisd-succifb nj

I

X2,

p. :;=,

Egcci


8/16

?20 G. J . O N \ F G E D W _ , A\ .

/ ,

@,

ifb

f

lg is df t`g prnnj nj (9). Ly [rnpnsdtdnf


9/16

O @ I \ I O R G \ D Q I R D N F NJ I OEI__ NJ RN\_DNF

J \ G G M \ N W [ _

?24

[rnnj. Is cgftdnfgb prgvdnusey, t`g gqudviegfog nj (9) ifb (


10/16

?7;

G . J .O N \ F G E D W _ ,A\ .

[rnnj.

R`it M( )ds `ncnmgfgnus if b quisd sgpirileg jneenws

jrnc R`gnrgc ?.9,Egcci


11/16

O@ I \I ORG\D QI RD N F

NJIOEI__NJRN\_DNF

J \ G G M \ N W [ _

?79

t`gf

@ ds

oncpegtgey bgoncpnsileg.

2* Ippedoitdnfs tn sgpirileg mrnups* R`gnrgc ?.9 oif lg

gxtgfbgb ienfm t`gedfgs nj R`gnrgc ?.7 tn bgsordlg i oeissnj

sgpirileg mrnups

df

w`do` gio` purg sulmrnup

nj

jdfdtg rifh

ds i

quisd succifb.

Df

nrbgr

tn

utdedzg

i

t`gnrgc

nj \.

Ligr X9,

p . 99=,

R ` . 99.?S,

wg

fggb

i

sgrdgs

nj

prgedcdfiry rgsuets.

]g

onftdfug

tn

onfstruot jufotdnfs w`do` dfbuog bgoncpnsdtdnfs.

\goiee

X9, p. 0;, Bgj. 7.9S

t`it

i g Mds

prdcdtdvg

nj

typg

dj

t

M

(i)3 ifb`

M

{i)3

M{ )

(i+

M *( ) ).

I

jdfdtg sulsgt

njMds

prdcd

tdvg

dj dts

gegcgfts

irg

prdcdtdvg wdt` bdjjgrgft typgs.

EGCCI 2.9.EgtJ(M)lg

()

@' + O

X( )

nftn

,sn@ds i

bdrgot succifb.

EGCCI 2.


12/16

?7


13/16

O @ I \ I O R G \ D Q I R D N F

NJ I

OEI__

NJ

RN\_DNF

J \ G G M \ NW [ _ ?7?

o n f b d t d n f ,

ufedhg oìdfs nj `gdm`ts. Jnr t`gsihg njbgjtfdtgfgss1 ly

i `gdm`t wgcgif i jufotdnf jrnc t` g sgt nj ieeprdcgs tn t` g sgt

nj fnffgmitdvg dftgmgrs ufdnf nn

ly i

typg

wg

cgif

if

gqudviegfog

oeiss

nj

`gdm`ts ufbgr

t`g

stifbirb gqudviegfog rgeitdnf

nf

`gdm`ts

X= , 07S8 ly i rgprgsgftitdnf nj i typg wg cgif i o`ndog nj i

`gdm`t `g .

EGCCI 7.9. Gvgry

onuftileg

oìdf nj typgs oif lgrgprgsgftgb

ly i oìdf nj `gdm`ts.

[rnnj. Egt

j

characterization of a class of torsion free groups in terms of endomorphisms

Documents