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INFINITE ABELIAN GROUPS WHITEHEAD PROBLEM

AND SOME CONSTRUCTIONS

BY

S H R O N S H E L H

BSTR CT

W e solve here som e problems from Fuchs book. W e show that the answer to

Whitehead s problem (for groups of power ~1) is independent from the usual

axioms o f set theory. W e prove the e xistence of large rigid system s for grou ps

of power 2, with no restr ict ion on 2. W e also prove that there are m any non-

isomorphic redu ced sep arable p-groups.

1 Introduct ion

H e r e a g r o u p m e a n s a b e l i a n g r o u p .

W h i t e h e a d s p r o b l e m ( se e , f o r e x a m p l e , [8 , P r o b . 7 9 , p . 1 8 4 ] ) is f o r w h i c h g r o u p

G d o e s E x t ( G , Z ) = 0 h o l d . I n o t h e r w o r d s , i f h : H ~ G i s a n e p i m o r p h i s m w i t h

k e r n e l Z ( th e in t e g e r s ) t h e n H c a n b e r e c o n s t r u c t e d f r o m G i n o n e w a y o n l y : a s a

d i r e c t s u m . M o r e p r e c i s e ly , t h e r e is a h o m o m o r p h i s m g : G ~ H , h g = 1 ~. S u c h

g r o u p s a r e c a ll e d W - g r o u p s ( W h i t e h e a d g r o u p s ) . B y S te in [ 2 2 ] a n d R o t m a n [ 1 9 ]

( o r , f o r e x a m p l e , [ 8 , 9 9 . 1 ] ) e a c h W - g r o u p i s N l - f r e e a n d s e p a r a b l e . I n p a r t i c u l a r

W - g r o u p s a r e t o r s io n - f r e e , a n d f re e g r o u p s a r e W - g r o u p s . H e n c e a c o u n t a b l e

g r o u p i s a W - g r o u p i f f i t i s f r e e . F o r a n ( i n f in i te ) c a r d i n a l 2 l e t :

( W a ) : e a c h W - g r o u p o f p o w e r 2 i s f r ee .

W e p r o v e i n S e c t io n 3 th e i n d e p e n d e n c e o f (W ~ I ) f r o m t h e u s u a l a x i o m s o f s e t

t h e o r y ( Z F C : Z e r m e l o - F r e n k e l w i t h t h e a x i o m o f c h oi c e) . B u t w e d o n o t u s e th e

m e t h o d s o f C o h e n [ 1 ] d i re c t ly . R a t h e r w e r e ly o n p r e v i o u s i n d e p e n d e n c e p r o o f s ;

t h a t i s , v a r i o u s a d d i t i o n a l a x i o m s h a v e b e e n s h o w n t o b e c o n s i s t e n t w i t h Z F C

( a s su m i n g t h e c o n s i st e n c y o f Z F C ) . N o w i f Z F C + X i s c o n s i st e n t a n d f r o m i t w e

Received Septemb er 4, 1973

2 4 3

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2 4 4 s . S H E L A H I sr a el J . M a t h .

can p ro v e a s t a t em en t X ' , t h en o f co u r s e

X

i s a l so cons i s ten t wi th ZFC. Thus we

prove tha t i f V = L then (W ~) . (W e cou ld p rove , in fac t, tha t (Wz) wi th 2 < No,~

ho lds . The d i f ficu lty in ~o ,, seems to me to be in L . ) O n the o ther hand , we p rove

th a t i f M A (M ar t i n ax io m ) an d 2 > Nt th en (W ~ ) f a il s. T h u s b y Gt id el [1 1 ] an d

M ar t i n an d So lo v ay [1 7 ] , (W ~ ) i s i n d ep en d en t o f Z FC . T h e q u es t i o n o n t h e i n-

d ep en d en ce o f (W s , ) f ro m Z F C + 2 = N~ rem a in s o p en . (As V = L im p l ie s

G C H , the con s i s tency i s c lear .) I f V = L , then every W -group i s f ree .

Fu ch s [8 , P ro b . 5 2, p. 5 5 ] a s k ed fo r t h e n u m b e r o f n o n - i s o m o rp h i c s ep a rab l e

p-g roup s o f card ina l i ty 2 . The an swer i s tha t the num ber i s 2 a. Fo r 2 = No th i s is

imm edia te , and fo r 2 > No i t fo l lows , in fac t , f rom a resu l t o f the au tho r on the

n u m b er o f n o n - i so m o rp h i c m o d e l s o f a n o n - s u p e r s tab l e t h eo ry (p a r t o f i t ap p ea r s

in [20 , Sec t. 3 ] ) . In Sec t ion 1 we g ive the p r oo f fo r regu lar 2 ; as fo r s ingu lar

ca rd in a l s , t h e p r o o f is co m p l i ca ted . No te t h a t b y t h e co n s t ru c t i o n o f (1 .1 ) we can

s h o w th a t i f V = L t h en , b y J en sen [1 4 ], fo r ev e ry r eg u l a r n o n -weak ly -co m p ac t

card ina l 2 , there i s a separab le p -g roup o f pow er 2 which i s no t the d i rec t sum of

cy c l ic g ro u p s , b u t ev e ry s u b g ro u p o f s m a l l er c a rd in a li t y i s. (Fo r weak ly co m p ac t

card ina l s there i s no such g roup ; weak ly compact card ina l s a re inaccess ib le and

rare. ) This part ia l ly answ ers [8 , Pro b. 56, p . 55,1 . (Inde pen dent ly , M ekl er [161,

Ek lo f [2,1 and Gre gory ob ta ined s imi la r resu lt s , and E k lo f [3 ] , Gre gory [12 ,1

ob tain ed bet te r resul ts . See also E kl of [4 ,1 .

T h e p ro o f i n d ica t e s t o m e t h a t s ep a rab l e p -g ro u p s can n o t b e ch arac t e ri zed b y

any reaso nab le se t o f invar ian ts . (Th is answers Prob lem 51 o f Fuchs [8 ] . ) Because ,

f i rst of al l , (us ing (1.2) notat io n), in o rder to cha racterize G(B) , we n eed B/D(o~I)

and then i f V = L, as in (3 .4) , we can define G ( B ) in the same way , us ing on ly

different r /6 's , and ob tain

G (B) ~ G(B) .

A r ig id sys tem of g roup s i s a fami ly wi th on ly tr iv ia l hom om orph ism s be tween

i t s mem bers ( i f h : G - -, H then h = 0 o r G = H , h ( x ) = r x fo r some ra t iona l r ) .

Fuchs , wi th the he lp o f Corner , p roves induct ive ly tha t fo r every 2 smal le r than

the f ir s t inaccess ib le card ina l , there i s a r ig id sys tem o f 24 to rs ion-f ree g roup s o f

ca rd in a l it y 2 . In S ec t io n 2 we r em o v e th e r e s t ri c ti o n o n 2 an d o u r p ro o f d o es n o t

u s e in d u c t i o n . N o te t h a t e ach m em b er o f a r i gid s y s t em is i n d eco m p o s ab l e . T h i s

answers P rob lem 21 f rom [9 , p . 183]. Fuch s [10 ,1 succeeded in rep lac ing the f i r st

inaccess ib le card ina l b y much h igher icard ina ls in an induct ive p roof . F uchs k ind ly

draw s m y a t ten t ion to the fac t tha t Th . 2 .1 a l so so lved Prob lem 37 in [7 , p . 208 ,1 ,

t h a t i s, t h e re a re 2 a n o n - i s o m o rp h i c co m p ac t an d co n n ec t ed g ro u p s o fca rd in a l i t y 2 a ;

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Vol . 18, 1974 W HITE HE AD PROBLEM 245

w i th s o m e c a r e w e c a n m a k e t h e m a l g e b r a ic a l ly i s o m o r p h i c . F o r t h i s u se t h e d u a l i t y

t h e o r e m o f P o n t r y a g i n , H e w i t t, a n d R o s s [ 1 3 ] .

I n a f o r t h c o m i n g p a p e r w e s h a ll p r o v e t h a t t h e r e a r e e s se n t ia l ly i n d e c o m p o s a b l e

p - g r o u p s o f a r b i t r a r i l y la r g e c a r d i n a l i ti e s ( a n s w e r i n g p o s i t iv e l y a q u e s t i o n o f P i e r c e

[ 1 8 ] r e p e a t e d i n [ 8 , P r o b . 5 5 , p . 5 5 ] ) . W e s h a l l a ls o p r o v e t h a t f o r a r b i t r a r i l y l a r g e

c a r d i n a l i t y 2 t h e r e is a s y s te m o f 24 p - g r o u p s s u c h t h a t t h e h o m o m o r p h i s m s b e t -

w e e n d i f f e r e n t m e m b e r s a r e s m a l l ( f o r d e f i n i t i o n s e e [ 7 , ( 4 6 , 3 ) , p . 1 9 5 ] ) . ( T h i s

a n s w e r s p o s i t i v e l y P r o b l e m 5 3 o f [ 8 , p . 5 5 ] .) A n o t h e r c o n s t r u c t i o n g i v e s f o r

/ t = 2 ~ = 2 ~ > 2 ~ a f a m i l y o f 2 s e p a r a b l e p - g r o u p s o f p o w e r /~, s o t h a t a n y

h o m o m o r p h i s m b e t w e e n d i ff e re n t m e m b e r s h a s ra n g e o f c a r d i n a l it y < 2 .

W e a s s u m e k n o w l e d g e in n a i v e s et t h e o r y , a n d i n s e p a r a b l e p - g r o u p s a s in

[ 7 , V I I

;[8 XI].

N O TA TIO N . L e t 2 , ~ , x d e n o t e i n f i n i te c a r d i n a l s , 0t,

f l ,? ,5 , i , j

o r d i n a l s , 6 a

l i m i t o r d i n a l ,

k, l , m, n , M , N

n a t u r a l n u m b e r s o r i n t e g e r s , to t h e f i r s t i n f i n i te

ord ina l . W e l e t r/, z , v be sequ enc es o f o rd ina l s . Le t l (r /) be t h e l en g th o f r/, r /( i) i t s

i t h e l e m e n t . L e t c f [ ~ ] b e t h e c o f i n a l it y o f ~.

G,H,

a n d s o m e t i m e s K , I , R a r e g r o u p s , h , g a r e h o m o m o r p h i s m s , p , q a r e p r i m e

n a t u r a l n u m b e r s , r a r a t i o n a l o r s o m e t i m e s a p - a d i c i n t e g e r .

W h e n n o t a t i o n b e c o m e s c o m p l e x , a i ( j ) i s w r i t t e n a s

a[j, i], al

a s

a[i] .

1 . T h e r e a r e m a n y s e p a r a b l e p g r o u l r s

H e r e a g r o u p m e a n s a r e d u c e d s e p a r a b l e p - g r o u p , t h a t i s, a g r o u p G s u c h t h a t f o r

e v e r y a 6 G (a :/: 0 ) f o r s o m e

n, p a

= 0 , a n d f o r s o m e n n o b E G sa t is f ie s a =

p b.

D EF IN IT IO N 1 .1 . F o r e v e r y l i m i t o r d i n a l g o f c o f i n a l i t y b i g g e r t h a n t o,

( a ) l e t D(g) be t he f i l t e r o f subse t s o f c( = { fl : f l < c(} ge ne ra t e d b y t he c los ed

u n b o u n d e d s u b s e ts o f c(;

(b ) we wr i t e A c_ B [m o d D (~ t) ] i f a - - [A - B] E D(ct) , s im i l a r l y fo r A = B ,

A : ~ B [ m o d D ( a ) ] ;

( c ) A _ a i s ca l l e d a

s ta t ionary

( sub se t o f a ) i f A ~ - 0 [m o d D(ct) ].

THEOREM 1 .1 (See So lov ay [2 l ] ) .

I f2 i s a regu lar card ina l b igger than

No, A , a

s tat ionary subset o f 2 , then A can be part i t ioned in to 2 pairw ise dis joint s ta t ion-

ar y subsets of 2.

R EM A RK . T h e p a r t i c u l a r c a s es w e n e e d c a n b e p r o v e n m o r e e a s i ly .

THEOREM 1.2.

I f

2 > N O

is a regu lar card ina l , then there i s a fa m i ly o f 2 x

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246 s. SHE LA H Israel J . M ath.

n o n - i so m o r p h i c ( r ed u ced ) s ep a r a b l e p - g r o u p s , each o f ca r d i n a l i t y 2 . M o r eo ver

no one o f t hem i s i somorph ic t o a subgroup o f t he o ther .

P R OO F . I t i s w e l l k n o w n t h a t A = { ~ :0t < 2 , ~ h a s c o f i n a l i t y N o } i s a s t a t i o n a r y

s u b s e t o f 2. B y ( 1 .1 ) w e c a n h a v e A = U ~ 6 ; ~ B 2 . H e n c e

B ~ ~ C _q B ~ t3 C , s o ( * ) h o l d s , c o m p l e t i n g t h e p r o o f .

2 . R i g i d s y s t e m s

Tr lEOgEM 2 .1 .

F o r a n y 2 t h e r e is a f a m i l y

{Gl : i < 2 x}

of torsion-free gr ou ps

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V o l. I 8 , 1 97 4 W H I T E H E A D P R O B L E M 2 4 7

e a c h o f p o w e r 2 , w h i c h i s a r i g i d s y s t e m , t h a t i s, i f h : G , --* G 3 i s a n o n - z e r o

h o m o m o r p h i s m , t h e n i = j a n d h ( x ) = n x f o r s o m e i n te g e r n .

P R OO F . W e r e s t r ic t o u r s e lv e s t o 2 > N o r e g u la r 9 L e t a l l t h e p r i m e s m e n t i o n e d

b e l o w b e d i st in c t . L e t a i ~ , c~< co + c o, i < 2 b e f r e e g e n e r a t o r s o f a g r o u p G . L e t

A b e a s u b s e t o f 2 . W e n o w d e f in e s o m e e q u i v a l e n c e r e l a t i o n s o v e r s u b se t s o f

{ a~ : c~ < c o + c o, i < 2 } (w e d e f i n e t h e m b y a g e n e r a t i n g s e t o f p a i r s ) :

( 1 ) E ~ = { ( a i ~ : i < ).} f o r c~ < co + c o. N o t i c e e a c h e q u i v a l e n c e c la s s h a s a t

m o s t t w o e l e m e n t s .

(2 ) E ~ : ( 0 < n < co) r e n a m e {a~ ' : i < 9 .} as {b~', :

i , j

< 2 } , a n d l e t E ~ b e

g e n e r a te d b y { ( a ~ b i * j ) : i < ,; ., j < 2}

( 3 ) E 2 : ( 0 < n < c o) i s g e n e r a t e d b y { ( a ~ b ~ .,i ) : i , j < 2 }

( 4 ) E~ : ( 0 < n < c o) f o r e a c h t5 < ; t o f c o f i n a l i t y N O c h o o s e a n i n c r e a s i n g

s e q u e n c e r/~ o f o r d i n a l s o f l e n g t h c o w h o s e l i m i t i s 6 . L e t E n a b e g e n e r a t e d b y

{ ( a ~ b~cn),~ : di < 2 , c f 6 = c o} . N o t i c e t h a t e v e r y e q u i v a l e n c e c l a s s h a s a t m o s t

t w o e l e m e n t s .

( 5 ) E 4 : B y T h e o r e m 1 .1 w e c a n f i n d ). d i s j o i n t s t a t i o n a r y s u b s e t s J i , i < 2 o f

{ ~ < ) .: cfc 5 = 0 9 } . L e t E o* b e g e n e r a t e d b y { ( a ~ : j ~ d , } .

( 6 ) E ~ : 0 < n < o~ r e n a m e { a~ + n : i < 2 } a s { b , : z a d e c r e a s i n g s e q u e n c e o f

o r d i n a l s l es s t h a n 2 o f l e n g t h n } i d e n t i f y i n g a i ~ '+ r a n d b~i~ . E n Sis g e n e r a t e d b y

{ ( b , , b , ^ a ~ ) :

l ( z ) = n , i < 2 , " r ^ ( i )

i s d e c r e a s i n g }

( 7 ) E~o : i s g e n e r a t e d b y { (a ~ ,~

a l o ) : i , j e A }

( r e m e m b e r A w a s a n a r b i t r a r y s u b -

s e t o f 2 } .

L e t J ~ ' = {c5 < 2 : c f 6 = c o , f o r s o m e i < 2 , a J t

b j , } . W e c a n a ss u m e J ~ n J~

i s s t a t i o n a r y f o r a n y i , j .

N o w w e c a n d e fi n e a g r o u p

G ( A )

c o n t a i n i n g G , w h i c h is c o n t a i n e d i n t h e d i v i s i b l e

h u l l o f G . G ( A ) i s g e n e r a t e d b y

a ) 6 ;

( b ) ( p , , ) - l a ~ ( f o r a n y 0 < l < o 9, ~ < c o + o 9, i < 2 ) ; a n d

( c ) ( p ~ ) - ' ( a ~ - a ] ) ( f o r a n y 0 < l < 0 9, w h e n a ~ E ~ a ~ ) ( o f c o u r s e p ~ , p ~ a r e d i s -

t i n c t p r i m e s ) .

W e s a y x e G ( A ) i s d i v i s i b le b y p ~ i f f o r a n y l < co f o r s o m e y ~ G ( A ) , p ~ y = x .

N o w n o t ic e th a t :

( * ) i f x = ~ , r~ a ~ e G ( A ) ( c l e a r l y o n l y f i n i te l y m a n y r 7 a r e :# 0 ) t h e n

(1 ) x i s d i v i s i b l e b y (p~) oo i f f f l # c t ~ r /a = 0 ,

( 2 ) x i s d i v i s i b l e b y ( p ~ 0 i f f f o r e a c h a ] E ( r T " ~ " p

9a l E ~ a j } i s z e r o .

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2 4 8 S . S H E L A H I sr ae l J . M a t h .

E x c e p t f o r p = p o , t h e d i v i s i b i l i t y b y p ~ i n G ( A ) , G ( B ) d o e s n o t d e p e n d o n t h e

c h o i c e o f A , s o w e d o n ' t m e n t i o n i t .

I t s u ff ic e s t o p r o v e t h a t { G ( A , ) : i < 2 4} i s t h e r e q u i r e d f a m i l y , w h e r e { A i : i < 2 ~}

i s a f a m i l y o f s u b s e t s o f 2 , n o o n e c o n t a i n e d i n t h e o t h e r . S o i t s u ff ic e s t o p r o v e t h e

f o l l o w i n g .

S u p p o s e h : G ( A ) -~ G ( B ) i s a n o n - z e r o h o m o m o r p h i s m ; t h e n A ___ B , a n d f o r

s o m e i n t e g e r n , h ( x ) = n x f o r e v e r y x ~ G ( A ) .

OBSERVATION I . h ( a ~ ) i s a l i n e a r c o m b i n a t i o n o f { a T : j < 2 } w i t h r a t i o n a l

c o e f f ic i e n t s. T h i s i s b e c a u s e a ~, h e n c e

h ( a ~ )

i s d i v i s i b l e b y p ~ , a n d b y ( * 1 ).

O BS ER VA TIO N I I . j o = { 3 < 2 : c f 3 = co , h ( a ~ v~ 0 } is s ta t i o n a r y a n d i n c l u d e s

J ~' f o r s o m e j .

A s h # 0 a n d G ( B ) i s t o r s i o n f r e e , h ( a ~ ) 0 f o r s o m e ~ , j . A s ~ ,j ~ ~ ,j,

o _ a ~ h e n c e i t d iv i d e s h ( a ~ - h ( a ~ ) , b u t b y O b s e r v a t i o n I a n d ( * 2 ) ,

i v i d e s a j

(p ~ )O o d o e s n o t d i v i d e h ( a ~ ) , h e n c e h ( a ~ ) O . S i m i l a r l y ( p~ )O o d i v i d e s

O ]

h ( a ~ ) - h ( b ~ a ) f o r a n y i < 2 , b u t b y O b s e r v a t i o n I a n d ( * 2 ) i t d o e s n o t d i v i d e h ( a j

0 0

( a s f o r ~ # ~ < ) . n o t a ~ E 1 a ~ ) h e n c e h(b ~ i ) : O. A s b e f o r e a ~ 1

= by .i => h(al~ ) 0

~ . h ( a ~ r 0 h e n c e b y t h e d e f i n i t i o n o f J t * w e a r e f i n is h e d .

O n S ER V A T tO N l l I . I f / ( z ) = n , l e t h ( b , ) = r 1 b ,~ + . . . + r i b , , ( r I O , . . - , r z = 0 ) ,

l ( z x ) . . . . . l ( z 3 = n ( b y O b s e r v a t i o n I ) ; t h e n z t ( n - 1 ) , - , z t ( n - 1 ) > z ( n - 1 ).

( M a y b e l = 0 . )

W e p r o v e t h i s a s s e r t i o n b y i n d u c t i o n o n z ( n - 1 ). I f z ( n - 1 ) i s z e r o , i t i s i m -

m e d i a t e . S o s u p p o s e z ( n - 1) = 7 + 1 ; l e t v = z ^ ( 7 ) , h ( b ~ ) = r l b , ~ + . . . + r m b v ~

( r ~ O, l (Vk) = n + 1 ). N o w ( p S ) ~ d i v id e s h ( b , ) - h ( b , ) , s o b y ( * 2 ) a n d t h e w a y E ,

w a s d e f i n e d f o r e v e r y j , 1 =< j < l , t h e r e i s a n i , 1 _< i _< m , s u c h t h a t z j i s a n

i n i t i a l s e g m e n t o f vi. A s vi i s d e c r e a s i n g z i ( n - 1) > v , ( n ) > v ( n ) = ? ( u s i n g t h e

n d u c t i o n h y p o t h e s i s ) , h e n c e z j (n - 1 ) > ], + 1 = z ( n - 1 ). S o w e p r o v e d O b s e r -

v a t i o n I I I . I f z ( n - 1 ) i s a l i m i t o r d i n a l , t h e p r o o f is s i m i l a r .

O BS ER VA TIO N 1 V . T h e r e i s a c l o s e d u n b o u n d e d s e t C ~ 2 s u c h t h a t i f ce < f l,

h a o

= r l a ~ ( 1 ) + . .. + r , a ~ , ) ( r i O ) t h e n ~ < ~ ( i ) < ,3 . ( M a y b e n = 0 )

o ,~ +1 o b (,> , a n d t h e d e f i n i t i o n o f o

s ( p ~ 1 7 6 d i v i d e s a , a , a ~

- = - -

E o + l , n e c e s -

e ~ + l _ ~ + 1 .

s a r i l y h ( a ~ + 1 ) i s r t a ~ ( l ) + . . . + r , a , ( , ) , h e n c e cz < a ( l ) b y t h e p r e v i o u s

o b s e r v a t i o n . L e t f i ( ~ ) = m a x { ~ ( l ) + 1 : l }, f 2 ( a ) = s u p { f l ( ~ : B < ~ } , a s 2 i s

r e g u l a r c~ < 2 : ~ f 2 ( ~ ) < 2 , a n d a s i n a d d i t i o n 2 > N o , C = { ~ : f 2 ( ~ ) = ~ } i s

c l o s e d a n d u n b o u n d e d .

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V o l . 1 8, 1 97 4 W H I T E H E A D P R O B L E M

2 4 9

O B S E RV A n O N V . I f 6 ~ C , o f 6 = c o, t h e n h a ~ = r 6a ~ f o r s o m e r 6. S u p p o s e

h a o ) o o (r~ ~ 0 ). B y O b s e r v a t i o n I V , w e c a n a s s u m e

= r I a i x ) -a t- . . . + r n a i n )

6 _~ i (1 ) = i x < . . . < i n ) = i . < 2 . A s s u m e i . > 6 . B y t h e d e f i n i t i o n o f t h e r h s

w e c a n c h o o s e m < to s u c h t h a t i f c f [ i l + x ] = to r h , + , (m ) > i~, a n d i f il > 6 ,

c f r q ] c o

t h e n q , , ( m ) > 6 . N o w ( p a ) ~ d i v i d e s a ~ - b m

t r a m , 6 , h e n c e

h a ~ - h ( b~ , m ~ ,~ ) . B y t h e d e f i n i t i o n o f E ~ , n e c e s s a r i l y i ( 1 ) , . . . , i n ) h a v e c o f i n a l -

i t y c o , a n d

h(bvtm.~],~) = r x b n p . , i t l ) l . i o ) + . . . + r m b v t ~ , i . ) ] , i t , ) .

A l s o ( p~ )~ o d i v i d e s o m .~ tm ,~ - b ~ t~ m , 6, h e n c e n e c e s s a r i l y b y ( * 2 ) a n d O b s e r v a t i o n I

( a s t h e r / [ m , i ( / ) ] ( 1 < I < n ) a r e d i s t i n c t ) , h a ~ , . , ~ ] ) o o

r l a ~ [ m , i l ) ] - . J - .. . .. j _ r n a n [ r a . l n ) ] .

T h i s c o n t r a d i c t s O b s e r v a t i o n I V . S o

e i t h e r

i . = 6 s o n = 1 a n d t h e n

h a ~ = r z a ~

o r n = 0 a n d t h e n c h o o s e r 6 = 0 .

O aS ER V A TIO N V I . F o r e v e r y i , h a ~ = r a i ~ f o r s o m e r . U s i n g p 0 w e s ee t h a t

f o r 6 ~ C , c f 6 = c o ; h a 6 1 ) = r ~ a ~ . N o w f o r e v e r y i < ) . , t h e r e i s a 6 ~ C c ~ J l ( a s J l

i s s t a t i o n a r y ) s o c f 6 = co . N o w ( p 4 ) ~ d i v i d e s a ~

-a2 so

i t d i v i d e s h a ~ - r , a ~ ,

s o b y O b s e r v a t i o n I a n d t h e d e fi n i t io n o f E ~ a n d ( * 2 ),

h a i ) = r 6 ai~

S o f o r e v e r y

i , h a ~ = r , a ~

N o t i c e t h a t 6 e J ~ n C i m p l i e s r~ = r 6 , h e n c e f o r 6 ( 1 ) , 6 ( 2 ) e J ~ n C , r~ < x) = r6 ~2 ).

F o r a n y i , c o n s id e r t h e h o m o m o r p h i s m

h i , h x x ) = h x ) - r ix .

I f h a = 0 w e f in i s h ,

o t h e r w i s e a l l o u r o b s e r v a t i o n s c a n a p p l y t o i t. A s J * r~ J~ i s s t a t i o n a r y f o r a n y i , j ,

w e o b t a i n a c o n t r a d i c t i o n t o O b s e r v a t i o n I I . S o r i = r . B y E ~ w e s ee th a t

h a ~ ) = r a 7 .

S o n o w w e c a n f in i sh t h e p r o o f . C l e a r l y

h x ) = r x

f o r e v e r y

x e G A ) .

T h e

r a t i o n a l r s h o u l d b e a n i n t e g e r , a s f o r e v e r y p t h e r e is a n

x e G A )

s u c h t h a t

x / p q~ G A ) . U s i n g E o c l e a r l y A _ B .

Q t m s a- io N . I s t h e r e a c l a s s o f t o r s i o n - f r e e g r o u p s , h a v i n g 2 a m e m b e r s i n e a c h

c a r d i n a l 2 w h i c h i s a r i g i d s y s t e m ?

R EM A R K 2 . 1 . T h e c o m p l e t i o n f o r s i n g u l a r c a r d i n a l s a n d N o i s e a s y . I f 2 i s s i n -

g u l a r , ~. = ] ~ ~

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250 S, SHEL AH Israel J . M ath.,

3 . W h i t e h e a d p r o b l e m

R e c a l l t h a t a g r o u p G i s c a l le d a W - g r o u p ( W h i t e h e a d g r o u p ) i f f o r e v e ry

h o m o m o r p h i s m h : H --, G o n t o G w i t h k e r n e l Z ( th e in t eg e r s) , t h e r e i s a h o m o m o r -

p h i s m 9 : G -* H , s u c h t h a t h g = 1a (s o H i s a d i r e c t s u m o f Z a n d a c o p y o f G).

W e d e a l w i t h a t o r si o n - fr e e g r o u p G o f c a r d i n a l i t y N t , s o w i t h o u t l o ss o f g e n e r a l i t y

i t s un iv e r se i s NI = w t = {~ :~ < cat}, an d G6 = {~ : ~ < 6} i s a pure su bg rou p

o f G . I t is k n o w n t h a t i f G i s a W - g r o u p , i t is s e p a r a b l e t o r s i o n f r e e a n d N1 - fr ee

( t h a t i s , e v e r y c o u n t a b l e s u b g r o u p i s f re e ). We c l a s s i fy t h e p o s s i b l e G to t h r e e p o s -

s ib i l i t i e s , bu t f i r s t we need a de f in i t ion .

D E , NInON 3 .1 . ( 1 ) I f

L ~ G, PC L, G)

i s t h e s m a l l e s t p u r e s u b g r o u p o f G

wh i c h c o n t a i n s L . N o t e t h a t i f H i s a p u r e s u b g r o u p o f G , L __. H t h e n

P C L , G)

= P C L , H ) .

W e o m i t G i f i t is c le a r .

(2 ) I f H i s a s u b g r o u p o f G , L a f i n i t e s u b s e t o f G , a ~ G , we s a y t h a t ~ ( a , L , H , G)

i f

P C H u L ) = P C H ) ~ P C L )

b u t f o r n o

b ~ P C H u L u

{a}) is

P C H

U L U {a}) =

P C H ) ~ P C L

U (b}) .

I ni}, i < su ch

OSSIBILITY I . F o r so m e 6 < ca1 th ere ar e a.[i~L~ = {a~ : I < cat,

t h a t :

(A) {a [ + G6 : i < ca t , l _~ n t} i s an ind ep en de nt f am i ly in

G/G6,

( B ) ( a .u ] , L i , G6 , G) ho l ds fo r an y i .

R EM A RK . W e c a n r e p l a c e G 0 b y a n y c o u n t a b l e s u b g r o u p o f G , a n d w . l .o . g .

6 ~ ]4/

POSSmILITY I I . N o t I , bu t the re a r e a s ta t io na ry se t A _ cat an d for an y

3 ~ A , L0 {a] : I < na}, n

an o)

s u c h t h a t :

(A) {a ] + G0 : l _~ n0} i s an in de pe nd en t f a m i ly in G/G,,

( B ) 0

zffa,(0) , L0, Go, G) ho ld s.

POSSmlHTY I I I . N e i th e r I no r I I .

R E M ARK. T h e c l a s si f ic a t i o n t o t h e t h r e e p o s s i b i li t ie s d e p e n d o n G o n l y u p t o

i s o m o r p h i s m . B e c a u s e i f h : G t ~ G 2 i s a n i s o m o r p h i s m , C = { 6 : h i s a n

i s o m o r p h i s m f r o m G ~ o n t o G f} i s cl o s ed a n d u n b o u n d e d s u b se t o f c a l. C l e a rl y C

i s c l o s e d , a n d i f ~ o < ca , d e f i n e b y i n d u c t i o n ~ , < c a 1, ~ , + t = s u p { h ( / ) : i < ~ , }

u {j : h j)

< a .} ] . Th en ~ ,+ t < cat a s c f ca t > No, an d s im i la r ly a* = U c~, < oJ t ,

~ * e c , s o C is u n b o u n d e d .

L e m m a 3.1. (1) Eac h pos sibi l i ty i s sat isf ied b y som e Nl- free group that i s,

every c ountable subg roup is f ree) . O f course, the poss ibi l it ies fo rm a part i t ion.

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(2) I f P o s s i b i l it y I f a i l s , t h en b y r e n a m i n g t h e e l em en t s o f G w e ca n a s s u m e:

(*) I f t~ < cot, {ch + G l: l ~_ n} i s ind epe nde nt , an d I t(an, L , Gp, G holds wh ere

L = { a l : l < n } , t h e n (a t + Gd+ w: l ~ n} i s no t i ndepend en t .

PR OO F. ( 1 ) F o r P o s s i b i l i t y I , l e t G b e g e n e r a t e d b y xn, n < co, an d

x~ = ~, ~=m (n /m )x~(n) f o r e a c h m < co , t / e C , w h e r e C i s a s e t o f N 1 i n c r e a s i n g

s e q u e n c e s o f n a t u r a l n u m b e r s o f l e n g t h co, s u c h t h a t ~ / z ~ C im p l i e s t h e r a n g e s

o f r/, ~ h a v e f in i te i n t e rs e c t i o n . ( T a k e G ~ t h e s u b g r o u p g e n e r a t e d b y t h e x n 's , n~ = 0

~ C = { ~ h : i < c o l } . )

nd ao = x~ , ,

F o r P o s s i b i l it y I I , c h o o s e f o r e a c h l im i t 6 < c ot a n i n c r e a s i n g s e q u e n c e r/a o f

o r d i n a l s o f l e n g t h c o , w h o s e l i m i t i s 6 . L e t G b e g e n e r a t e d b y x ~ , ~t < c o t, a n d

x"~ = Y~ ~= ,,(n /m )x ~t~,al (m < t~, 6 < co l , 6 a l im i t o rd ina l ) . (See [8 , (75 .1 ) ] . )

F o r P o s s i b i li t y ] I I ta k e t h e f r e e g r o u p w i t h N ~ g e n e r a t o r s .

( 2 ) W e d e f in e b y i n d u c t i o n a n i n c r e a s i n g s e q u e n c e o f l i m i t o r d i n a l s c t(i),

c t(0) = 0 , ct (6 ) = ~ t ~( i ) so th at a I ~ G~( i+l )

N o w r e n a m e { a :0 t(i) =< a < 0 c(i + 1 )} ( r e m e m b e r t h e e l e m e n t s o f G a r e o r d i n a l s )

b y

{f l :coi ~ f l < co(i+

1)}.

LEMMA 3 .1 (3 ) F o r N l - , f r e e G p o s s i b i li ty I I I is eq u i va l en t t o G b e i n g t h e

d i r ec t s u m o f co u n t a b l e g r o u p s .

P R O O F. C l e a r l y i f G = 0 )~ G , G~ c o u n t a b l e , t h e n w e c a n a s s u m e e a c h a ~ G ~

sa t is f i e s co~ =< a < co(i + 1 ), so Poss ib i l i t y I I I ho lds . Su pp ose Pos s ib i l i t y I I I h o l ds ;

t h e n P o s s i b i l i t y I f ai l s, h e n c e w e c a n a s s u m e ( * ) f r o m ( 2 ) . A s P o s s i b i l i ty I I f a i ls ,

w e c a n f i n d a c l o s e d a n d u n b o u n d e d s e t C _ { gi: 6 < c ot} s o t h a t f o r 6 e C w e

c a n n o t f in d a a n d f i n i te A s u c h t h a t

~(a, A , Go, G)

h o l d s . B y r e n a m i n g w e c a n

a s s u m e C = ( co l: i < c o a} . N o w f o r e a c h 6 , w e c a n f i n d G ; s o t h a t G ~ i s a p u r e

su bg ro up o f G, Ga+ 1 = I..J ~G~, G O = Ga an d

Ga" +I/G~

h a s ra n k o n e . N o w w e c a n

d e f in e b y i n d u c t i o n o n n > 1 , a , , ~ G a s o t h a t G , ~ = G a + P C ( a x , ' " , a , , ) . B y t h e

d e f i n i ti o n o f ~ th i s c l e a r ly c a n b e d o n e . L e t H 6 = P C ( a ~ , a 2 , . . . ) , so Ga+ ~ = Ga

H a h e n c e G = q ) a H a ( t ) G o , .

D~F~N~a'~Or~ 3.2 . A (G , Z )-g ro up i s a g ro u p H w ho se se t o f e le m en ts i s G x Z

= { ( a, b ) : a e G , b e Z } , a n d t h e m a p p i n g h : ( a , b ) ~ a i s a h o m o m o r p h i s m f r o m

H t o G w i t h k e r n e l Z = { 0} x Z ; a n d ( a , b ) + ( 0 , c ) = ( a , b + c ) . W e d e n o t e t h e

h c o r r e s p o n d i n g t o H i b y h i, a n d a ( G i, Z ) - g r o u p b y H I.

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L E ~ 4 A 3 .2 . ( 1 ) L e t U 1 b e a s u b g r o u p o f G 2, H I a ( G i , Z ) - g r o u p ; t h e n w e c a n

e x t e n d H 1 t o a ( G 2 , Z ) - g r o u p H 2 .

( 2 ) I f g l :G 1 ~ H 1 is a h o m o m o r p h i s m , h l g l = l o t : j a n d l r ( a ,A , G i , G 2 ),

w h e r e A i s f i n i t e , t h e n w e c a n e x t e n d H i t o a (U2, Z ) - g r o u p H 2 s o t h a t f o r n o

h o m o m o r p h i s m g 2

G2 -+ H2,

h292 l~r2~, a n d g 2 e x t e n d s g l .

P R O O F . T h i s i n f a c t i s n o t n e w .

( 1 ) T h i s is i m m e d i a t e b y 5 1 .3 ( 2) f r o m [ 7 , p . 2 1 8 ] . B u t w e n e e d ( 3 .2 ) o n l y f o r G 1

a W - g r o u p , s o w e p r o v e i t f o r t h i s c a se . B y it e r a t i n g , i t s u ff ic e s t o p r o v e t w o c a s e s :

( a ) G2/G 1 i s a f r e e g r o u p o f r a n k o n e , s a y g e n e r a t e d b y a o + G 1 . T h e n e v e r y

b l e G 2 h a s a u n i q u e r e p r e s e n t a t i o n a s nao + c (c 9 G1, n i n t e g e r ) . S o d e f i n e

(b 1, k l) Jr- (b 2, k2) = (n l ao + Cl; k l) d- (n 2 a 0 -}- c2, k2 )

d f

---- ( (n I -}- n2)a 0 + c3, k3)

wh ere ( i nH 1) ( c 1 , k l ) + ( c2 , k2) = ( c3 , k3) .

(b ) G2/G ~ i s a c y c l i c g r o u p o f a p r i m e o r d e r p , g e n e r a t e d b y a o + G x . S o d e f i n e ,

g l : G I - * H I a s a h o m o m o r p h i s m , h l g I = 1 ~ , a n d g x ( c) = ( c, re(c)) .

(n la o + c~ , ks) + (n2 a 0 + c2, k2) d f ( (n~ + n2 )a 0 + c I + c 2 , k I + k 2 - m(Cx)

- re(c2) + re(c1 + c2) + f ( n t + n 2))

w h e r e 0 ~_ n l , n 2 < p , a n d f ( n x , n 2 ) i s 0 w h e n n I + n 2 < p a n d M 9 o t h e r w i s e .

( 2 ) L e t A = { a x , -. ., a m } , G a = P C ( G 1 u A ) , G 5 = P C (G ~ w A u { a } ), a n d

G 4 t h e g r o u p g e n e r a t e d b y G 3 , a . N o w b y ( 1 ) e x t e n d H 1 t o a ( G 4 , Z ) - g r o u p H 4 . N o w

a n y h o m o m o r p h i s m g : G 4 - + H 4 e x t e n d i n g g l , h g g = I~E#;I i s d e t e r m i n e d b y t h e

v a l u e s o f g ( a i ) , g(a) , i = 1 , . . . , m , a n d a s g( a i) 9 ( (a i, l ) : l 9 Z } t h e r e a r c o n l y c o u n t -

a b l y m a n y s u c h g ' s : g ~, n < co . C l e a r l y G s / G 4 s h o u l d b e i n fi n it e s o e i t h e r

( A ) i t c o n t a i n s a n i n f in i te d i r e c t s u m o f c y c li c g r o u p s o f p r i m e o r d e r , o r

( B ) i t c o n t a i n s a c o p y o f Z ( p ~ ) f o r s o m e p .

C a s e A . S u p p o s e g e n e r a t o r s o f t h o s e g r o u p s a r c a~ + G 4 ( n < c o ), o f o r d e r p~ .

L e t G * b e g e n e r a t e d b y G 4 , a o , - , a n - I . W e d e f in e b y i n d u c t i o n ( G ~* , Z ) - g r o u p s

* I f H~* i s de f in ed , we de f ine H~*+ 1 as i n ( l b ) us in g a s

,, H*o= G~.,H , + t e x t e n d i n g H ~ .

c o n s t a n t M ~ , a n d t h e n b y ( 3 . 2 ( 1 )) f i n d a (G s , Z ) - g r o u p H s e x t e n d i n g a l l t h e H ~*. I f

g :G ~ -+ H 5 i s a h o m o m o r p h i s m , h g = l ~ts~ , t h e n f o r s o m e n , g e x t e n d s g ~ . L e t

p,a~ = b~ 9 G4, g,(b~ ) = (b~, k~), an d g(a~) = (c~ , l~ ) , (b~ , k~) = g(b~) = g(p~a~)

= p,g (a ~) = pn(a~ , l~ ) = (b~,p~l~ + M ,) s o k n = M ~ ( m o d p ~ ). H e n c e , i f w e c h o o s e

M ~ = k~ + 1 , H 5 sa t i s fi e s ou r r equ i r em en t s .

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C a s e B . S i m i l a r t o C a s e A ; h e r e p a n + t - a . = b . + t ~ G , , p a o = b o ~ G 4 a n d ,

l e t t ing

g b . ) = b . , I n ), g a . )

= a . , k . ) w e o b t a i n M o + p k o = 1 o, M n + 1 + p k n + l

= k n + I .+ t . S o f r o m t h e M . ' s w e c a n c o m p u t e t h e k . ' s , a n d f o r s u it a b l e M n ' s t h i s

i s im p o s s i b l e .

THEOREM 3.3 . A s s u m e C H , t h a t i s , 2 ~r176 N 1. T h e n i f G s a ti s f ie s P o s s i b i l i t y I ,

i t i s n o t a W - g r o u p .

R EM A RK . T h i s i s n o t r e a l l y n e e d e d f o r t h e i n d e p e n d e n c e o f ( W t , ,) .

P RO O F. W e s h a l l c o n s t r u c t a ( G , Z ) - g r o u p H . L e t H ~, b e t h e d i r e c t s u m o f G o,

a n d Z , a n d l e t {g i : i < c o l } b e t h e li s t o f a l l h o m o m o r p h i s m s f r o m G o, i n t o H ,o s u c h

t h a t

h g

= 1~ ( ex i s t s by

C H ) .

Le t G~ = G(c t ) =

P C [ G ~ , u

{ a / :

1 < n ( i ) ,

i < ~}],

a n d w e d e fi n e a ( G ~ , Z ) - g r o u p H ~ b y i n d u c t i o n o n ct. F o r ~ --- 0 , ct l i m i t t h e r e a r e n o

p r o b l e m s . I f w e h a v e d e f i n e d f o r H ~ , d e f i n e f o r H ~ + t s o t h a t g ~ c a n n o t b e e x t e n d e d

t o a h o m o m o r p h i s m g f ro m G * =

P C [ G o , W { a t : l

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25 4 S. SH EL AH Israel J . Math. ,

THEOREM 3.5 .

A s s u m e

M A + 2 ~ ~ > N 1

( M A f o r M a r t i n A x i o m ) . I f G h a s c ar -

d i n a l i t y N 1 , is N ~ - f re e , a n d d o e s n o t s a t i s f y P o s s i b i l i t y I t h e n i t i s a W - g r o u p .

R EM A RK . W e c a n e x t e n d t h e p r o o f t o g r o u p s o f c a r d i n a l i t y le s s t h a n 2 No

P R O O F . I f G s a t i s f i e s P o s s i b i l i t y I I I , t h e n b y ( 3 . 1 ( 4 ) ) i t i s t h e d i r e c t s u m o f

c o u n t a b l e g r o u p s . E a c h s u m m a n d i s f r e e b y t h e N l - f r e e n e s s , s o G i s f r e e , h e n c e a

W I - g r o u p .

S o a s s u m e G s at is f ie s P o s s i b i l i ty I I . S o l e t h : H -- , G b e a h o m o m o r p h i s m

w i t h k e r n e l Z , t h e s e t o f e le m e n t s o f H b e i n g

G x Z , h ( ( a , b ) ) = a .

M A s a y s t h a t

f o r a n y 2 < 2 s ~

( M A ~ ) : S u p p o s e P is a p a r t i a l o r d e r o f p o w e r =< 2 , a n d i n P t h e r e i s n o t a n y

s u b s et o f N ~ p a i r w i s e - c o n t r a d i c to r y e l e m e n t ( a , b a r e c o n t r a d i c t o r y i f t h e y h a v e n o

c o m m o n u p p e r b o u n d ) . S u p p o s e

{ D i : i

< 2 } a r e d e n s e s u b s e t s o f P ( t h a t is ,

( V i

< 2)(Va e

P ) ( 3 b ~ D ~) [a <

b] ) .

T h e n

t he re i s a se t G c p suc h t ha t G ~ D~

Z f o r a n y i < 2 , a n d a n y t w o m e m b e r s o f G h a v e a c o m m o n u p p e r b o u n d i n

17 ( suc h a 17 i s ca l l ed gen er i c ) .

L e t P b e t h e se t o f h o m o m o r p h i s m s # f r o m f in i te l y g e n e r a t e d p u r e s u b g r o u p s I

o f G i n to H , s u c h t h a t

h #

= 11. S o P h a s p o w e r N ~. T h e p a r t i a l o r d e r i s e x t e n d -

i n g , t h a t is , g l < g 2 i f f g 2 e x t e n d s g l . L e t f o r i < to a, D l =

{ g e P : i

i s i n t h e

d o m a i n o f g } . I f t h e r e i s a g e n e r i c 17 ~ P d e f i n e

g * ( x ) = y

i f f o r s o m e g E 17

g ( x ) = y .

A s

G n D l ~ ~ , g * ( i )

i s d e f i n e d a t l e a s t o n c e , f o r e a c h i < c o~ . A s a n y

t w o m e m b e r s o f G h a v e a c o m m o n u p p e r b o u n d , g * i s u n i q u e l y d e fi n ed , a n d is a

h o m o m o r p h i s m ; a l s o

h g *

= 1G , s o w e a r e f i n i s h e d . T h u s , i t s u f fi c es t o p r o v e :

( 1 ) E a c h D t i s d e n s e . T h i s is b e c a u s e I i s p u r e a n d f r e e l y g e n e r a t e d b y s o m e

a 1 , - . ., a ~ e G ; s o t h e r e i s a n a ~ . ~ s u c h t h a t a ~ , . . ., a ~ + ~ f r e e l y g e n e r a t e K f o r a n y

s u b g r o u p K o f G o f r a n k n + 1 w h i c h c o n t a in s I , f o r e x a m p l e ,

P C ( a 1 , . , a , , b ) .

( 2 ) S u p p o s e {g~ : i < t o 1 } __q P ; t h e g ~'s a r e p a i r w i s e c o n t r a d i c t o r y a n d w e s h a l l

o b t a i n a c o n t r a d i c t io n .

i , a~ . As we can r ep l ace {g i : i < t o1) by

e t D o m g i b e f re e l y g e n e r a t e d b y a 1,

a n y s u b f a m i l y o f t h e sa m e c a r d i n a l i t y t h u s w e c a n a s s u re n = n i. L e t { a l , , am }

b e a m a x i m a l s e t o f e l e m e n t s o f G , w h ic h g e n e r a te s f r e el y a p u r e s u b g r o u p a n d

{ a~ , . . ., a r,} -~ D o m g l f o r R l i 's . S o w i t h o u t l o s s o f g e n e r a l i t y

a l , . . ., a ,, ~ D o i n g

t

f o r e v e r y i, h e n c e w i t h o u t lo s s o f g e n e r a l i ty a ~ = a l , ' , am = am . A s th e n u m b e r

o f g t' s w i t h s o m e f ix e d d o m a i n is a t m o s t c o u n t a b l e , w e c a n a s s u m e D o m g ~

D o m g j f o r i ~ j ; h e n c e m < n , a n d a l s o { a l , ' , am } u ( a l : m < l < n , i < o~1}

i s i n d e p e n d e n t . N o w w e d e f i n e a s t r i c t l y i n c r e a s i n g s e q u e n c e o f o r d i n a l s c t( /) , i < a h ,

8/18/2019 Shelah 1974

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u 1 8, 1 9 74 W H I T E H E A D P R O B L E M 2 5 5

s u c h t h a t P C [ ( D o m g = (~ ) + G ~ , ] i s fi n i te l y g e n e r a t e d ( t h i s i s p o s s i b l e a s

i

P o s s i b i l i t y I f a i l s) . H e n c e f o r e v e r y i t h e r e a r e c ~ , - . . , c k ( o ~ G ~= s u c h t h a t

i

P C [ ( D o m g = ( O ) + Gt~,] i s g e n e r a t e d b y G t ,~ U { b l , , b k o )} , w h e r e { b ~ , . . . , }

~ - V C ( a ~ , . . . , ~ ~

n, c l , . . . , ck(i)) .

N o t i c e t h a t c~ < i o g. A s t h e s e t o f i < o91 s u c h t h a t io 9 = i i s c l o s e d a n d u n -

b o u n d e d w e c a n d e f in e a d e c r e a s i n g s e q u e n c e o f s t a t i o n a r y s e t s J o --- J ~ ~ ' ~ - j m

s u c h t h a t f o r i ~ J o , k ( i ) = m a n d f o r a l l i ~ J : , cz = c l lf o r s o m e cz ( b y [ 6 ] ) . N o w

f o r e a c h i ~ J , , w e c a n e x t e n d g = ~ ) , s o t h a t i ts d o m a i n ~ i

i l l b e

P C ( a 1 , . . . , a , , c D . . . , c , , )

a h d c a l l t h e n e w h o m o m o r p h i s m f . B y c h a n g in g n o t a t io n w e c a n a s su m e D o r a g~

l ' , a , ~ a n d { a l , . . ' , a m } t 3

{ a i : m < l < n ,

s a g a i n f r e e l y g e n e r a t e d b y a l , . . - , a m, a m + 1 ,

i e J m } i s i n d e p e n d e n t . B y r e p l a c i n g J m b y a s u b s e t o f t h e s a m e c a r d i n a l i t y J * w e

9 . , a ~ e G j .

a n a s s u m e f ( a l ) ( l = 1 , . . . , m ) a r e f ix e d . C h o o s e i < j ~ J * s o t h a t am+ 1,

B y c o n st r u ct io n , D o m # 1 + G j , o i s a p u r e s u b g r o u p o f G , s o D o r a g J + D o m # t i s a

p u r e s u b g r o u p o f G , a n d i ts d o m a i n i s f r e e l y g e n e r a t e d b y { a~ , . .- , am ,

I a ~ + l , a ~} . A s # ] ( a t ) g~(al) a c o m m o n e x t e n s i o n ( w h i c h b e l o n g s

m + l , . . . ,

an , . . . , =

t o P ) e x i s t s . C o n t r a d i c t i o n .

C O NC LU SIO N 3 .6 . T h e s t a t e m e n t E v e r y W - g r o u p o f c a r d i n a l i t y N t i s f r e e i s

i n d e p e n d e n t o f a n d c o n s is te n t w i th Z F C ( Z e r m e l o - F r e n k e l w i t h t h e a x i o m o f

c h o i c e , th e u s u a l s e t o f a x i o m s o f s et th e o r y ) . ( W e a s s u m e o f c o u r s e t h e c o n s i s te n c y

o f Z F C . )

P RO O F. B y S t e in [ 2 2 ] a n d R o t m a n [ 1 9 ] a n y W - g r o u p i s N l - f r e e a n d s e p a r a b l e .

B y G 6 d e l [ 1 1 ] , Z F C + V = L i s c o n s i s t e n t . B y ( 3 .4 ) , i f V = L a n y W - g r o u p

s a t is f i e s P o s s i b i l i t y I I I , s o b y ( 3 .1 ( 3 )) , G = ~ G l , G~ c o u n t a b l e . A s G i s N l - f r e e ,

G~ i s f r e e s o G i s f re e . T h u s Z F C + V = L i m p l i e s o u r s t a t e m e n t , h e n c e i t is c o n -

s i s t e n t .

B y M a r t i n a n d S o l o v a y [ 1 7 ] , Z F C + M A + 2 t% > N 1 i s c o n s is t en t . B y (3 .5 ) a n d

( 3 .1 ( 1 )) , Z F C + M A + 2 t% > N t i m p l i e s t h e e x i s t e n c e o f n o n - f r e e W - g r o u p s o f

c a r d i n a l i t y N ~ .

O P E N Q U E S TIO N . I s i t c o n s i s t e n t t h a t

( 1 ) e v e r y N t - f r e e g r o u p o f c a r d i n a l i t y N 1 is a W - g r o u p ?

( 2 ) t h e r e a r e s u c h g r o u p s , s a t i s f y i n g t h e s a m e p o s s i b i l i t y , o n e a W - g r o u p , t h e

o t h e r n o t ?

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256 S. SHELAH Israel J. Math.,

ACKNOWLEDGEMENT

I would l ike to th ank L. Fuchs and P. Ekl of for detecting many errors .

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IN~nrOTE OF MATHEMATICS

THE HEBREW UNIVERSITYOF JERUSALEM

JERUSALEM, ISRAEL