Symmetric Groups and the Open Sentence Problem

PATRAS LOGIC SYMPOSION GMetakidex (ed.) @North-Holland Publishing Cornpony. 1982

159

SYMMETRIC GROUPS AND THE OPEN SENTENCE PROBLEM Verena Huber-Dyson U n i v e r s i t y o f Calgary Calgary, A l b e r t a Canada

1.

INTRODUCTION The elementary t h e o r y o f a c l a s s

w

w i l l be denoted by TK.

and

K

o f structures o f f i x e d s i m i l a r i t y type

w i l l stand f o r the sets o f q u a n t i f i e r f r e e

3K

formulas whose u n i v e r s a l o r e x i s t e n t i a l c l o s u r e s b e l o n g t o

IQ

TK,

w h i l e we w r i t e

f o r t h e s e t o f t h o s e t h a t a r e s a t i s f i a b l e i n some K - s t r u c t u r e .

problem f o r

VK

i s what T a r s k i c a l l s t h e open sentence problem f o r

e q u i v a l e n t t o t h e d e c i s i o n problem f o r

The'decision

K.

t h e e q u a t i o n problem f o r

K3,

K, i n

Macintyre's terminology.

If

all finite

t h e n b o t h 'dK and Kf 3 a r e r e c u r s i v e l y enumerable, and

K-structures,

TI(

i s f i n i t e l y a x i o m a t i z a b l e and Kf

It i s

Xf 3 i s d i s j o i n t f r o m t h e s e t

consists o f

o f a l l n e g a t i o n s o f WK-formulas.

If K

is

closed under d i r e c t p r o d u c t s i t s e q u a t i o n problem reduces t o simultaneous s a t i s f i a b i l i t y of f i n i t e systems o f e q u a t i o n s t o g e t h e r w i t h one i n e q u a t i o n . Let

6, F

and B

s t a n d f o r t h e c l a s s e s o f a l l groups, o f a l l f i n i t e groups

and of a l l f i n i t e symmetric groups,

Sn, n

E

N,

denote t h e group of a l l permuta-

t i o n s o f f i n i t e s u p p o r t on a c o u n t a b l y i n f i n i t e s e t by c l a s s o f a l l i n f i n i t e models o f lem

TK.

Su,

and w r i t e

K,

for the

The u n d e c i d a b i l i t y o f t h e open sentence prob-

f o l l o w s f r o m t h e u n s o l v a b i l i t y o f t h e word problem f o r group t h e o r y .

In

1961, 111, Cobham proved t h e h e r e d i t a r y u n d e c i d a b i l i t y o f t h e t h e o r y o f f i n i t e symnetric groups and w i t h i t t h e u n d e c i d a b i l i t y o f f i r s t o r d e r f i n i t e group t h e o r y . I have been i n t r i g u e d by t h e open sentence problem f o r t h a t t h e o r y s i n c e 1963, [21. Since e v e r y f i n i t e group i s embeddable i n a s u f f i c i e n t l y l a r g e s y m n e t r i c one and

Su i s t h e u n i o n o f an ascending c h a i n o f c o p i e s o f t h e Sn's

we have

V. HUBERDYSON

160

More u s e f u l t h a n

So

i s i t s e x t e n s i o n by an automorphism t h a t c y c l i c a l l y p e r -

mutes a g e n e r a t i n g s e t o f t r a n s p o s i t i o n s . t r a n s p o s i t i o n and an i n f i n i t e c y c l e .

T h i s group

S

i s generated by a s i n g l e

T h a t i t s open t h e o r y c o i n c i d e s w i t h

\If: was

Here I s h a l l show how elementary a r i t h m e t i c can be d e f i n e d i n

proved i n [ 3 1 .

S

u s i n g e x i s t e n t i a l p r e d i c a t e s o f t h e language o f group t h e o r y e n r i c h e d by two constants for the generators.

From t h e u n s o l v a b i l i t y o f H i l b e r t ' s t e n t h problem, [51,

follows t h e u n d e c i d a b i l i t y o f t h e open sentence problem f o r S i n t h e extended language.

The f i r s t group t o which t h i s t y p e o f argument was a p p l i e d was t h e two [6].

g e n e r a t o r r e l a t i v e l y f r e e group o f n i l p o t e n c y c l a s s two,

Observe however t h a t

t h e open t h e o r y o f t h a t group i s a p r o p e r e x t e n s i o n o f t h e open t h e o r y o f n i l p o t e n t groups o f c l a s s two, which i s i n f a c t d e c i d a b l e , s i n c e i t i s a x i o m a t i z a b l e and n i l p o t e n t groups a r e r e s i d u a l l y f i n i t e .

The open sentence problem f o r t h e c l a s s o f

a l l n i l p o t e n t groups on t h e o t h e r hand i s s t i l l open t o o . Although I have n o t been a b l e t o e l i m i n a t e t h e two c o n s t a n t s w i t h o u t s p o i l i n g t h e s i m p l i c i t y o f t h e i n t e r p r e t a t i o n , t h e r e a r e a few i n t e r e s t i n g c o r o l l a r i e s , and a r e c o n s t r u c t i o n o f Cobham's r e s u l t , structure o f

S

[ l ] .I hope t h a t a deeper a n a l y s i s o f t h e

and of how i t encodes a r i t h m e t i c and t h e t h e o r y o f f i n i t e symmetric

groups w i l l improve t h e r e s u l t s below.

2.

THE GROUPS I t i s w i s e here t o d e a l w i t h c o n c r e t e groups r a t h e r t h a n w i t h isomorphism

types.

Our p e r m u t a t i o n s range o v e r t h e group

of t h e i n t e g e r s and



fi o f a l l b i j e c t i o n s on t h e s e t

stands f o r t h e subgroup o f

We s h a l l need t h e t r a n s p o s i t i o n

T =

(O,l),

for

i

E

2,

the basic cycles o f order

for

n

E

N,

t h e successor

p:i

H

n+l

generated by t h e s e t

i t s s p e c i a l conjugates 5, = (0,1,

i+land t h e groups

....n ) ,

c-,

T~

Z UC 0.

= (i,i+l),

= (0,-1.

...,-n),

161

Symmetric Groups and the Open Sentence Problem

The p e r t i n e n t p r o p e r t i e s o f these groups and permutations a r e c o l l e c t e d in the next t h r e e lemnas in an o r d e r t h a t leads r i g h t u p t o t h e lemnas of the next s e c t i o n . They a r e easy t o prove, [ 3 ] , by d i r e c t c a l c u l a t i o n s and inspection of diagrams. We use t h e abbreviation [x,y] f o r t h e commutator x-'y-'xy, the derived subgroup of c e n t r a l i z e r of

x

write U '

U generated by i t s commutators and w r i t e Z(G,x) W e c a l l an extension R '

in t h e group G.

for for the

o f a presentation

by a s e t of negations of equations complete i f t h e diagram o f th8 group



gp c o n s i s t s of consequences of

R'.

For b r e v i t y we indulge i n t h e abomination

o f concatenating equations. LEMMA 1.

(i)

S

i s the extension of

phism induced by the cyclic permutation (ii) S'

= S ,;

S' =

-

Su 'li

by its outer automorT

{[x,y] ]x,ycS}, S u = S ' U T S ' , and

normal subgroup of Sw (iii) z(s,p) = < p > , Z(S,pcil)

=

o f ~the generators,

~

+

S'

and of S, xSn, and Z(S,pkr)

(iv)

z(sn+l,c.n) = , Z(Sn+l,
(v)

every permutation in

=

is a mhimal = < PkT >

for k # 0,

< ckn ~ > ,for O
S has finite o r cofinite support and

finitely many orbits.

LEMMA 3. The following presentations hold for

T,

p,

a,b,c: S = gp, Sn+l

=

gp
5, interprerhg

,..., [a,cn-1I2,cn+l~t,)-n>.

V. HUBER-DYSON

162

The addition of the inequation [ a , b ] # l or [a,cl#l t o t h e above sets of relations makes the presentations complete.

3.

THE PREDICATES Extending t h e language

and

b

L

o f group t h e o r y by adding two c o n s t a n t symbols

t o t h e p r i m i t i v e s 1, * and

-'

we o b t a i n a language

L'

a

i n which we s h a l l

use t h e f o l l o w i n g terms and p r e d i c a t e s x'=b-'xba,

'x=bxa-lb-',

vO(x,y)'1, vk+l(x>Y)'(vk(x,y))' k=vk(a,b)=b- k (ba) k ,

I

vk,1(x3Y)"(vk(x>Y))

-

D(x,y) : [x,bl =[xy,bal= [ y , ~ - ~ a x ~[y,(xb)]= 1a x b l = l , C(x): (3z)D(z,x),

N(x) : (3z)D(x,z),

E(x ,y) : (3 z)(D(z,x)

& [y,x'b-ll = [ ~ , z - ~ a z ~ l = l ) ,

( 3 u ) ( 3 v ) ( D ( u S x ) & D(v,Y) & D(uv,z)),

Z(X,Y.Z):

: (3u) (3 v ) (3w) ( 3 r ) ( 3 s ) (3 t ) ( ( x = z = l ) v ( y = z = l )V(D(u ,x) & D(v,y)

n(x,y,z)

& D(w,z)

& D ( v 2 , r o ) & E ( r , s ) & t r , s l = [ v w t - 1s t , u b a l = l ) . The r e s t o f t h e p r e d i c a t e s belong t o

L

itself:

T ( x ) : x 2 =1 & ( W Z ) ( [ x , z ] = ~ ~ [ x , z ] ~ = ~ ~ [ x , z ] ~ = ~ ) , S(x,y):

T ( x ) & [x,y] 3 =1 & [ x , y ] # l & (Wz)( [ z , y l = l

G(x,Y):

S(X,Y) & ( W t ) - S ( t , y t ) ,

I(x,y):

(3t)(S(t,x)

D(x,Y):

I(Y,xY-'),

& S ( t i Y t ) & (Ws)(S(s,ts)

=> z=yVZ=y- 1V [ x , Z I 2 = l ) ,

& -S(s,ys)

& t s , y l # l => S(S,XS))),

Sm(x,y,z):

( 3 r ) ( 3 s ) ( 3 g ) ( I ( r - 1xr,s- 1y s ) & z - l = g - l z g = r - 1x r s - 1y s ) ,

Pr(x,y,z):

( 3 t ) ( 3 g ) ( 3 r ) ( 3 ~ ) ( 3 ~ ) ( 3 ~ ) ( 3 hk ))((S3( t , z ) & S(t,r-'xr)

8 S(t,s-'ys)

& O(z,r-'xr)

& z-'=g-lzg & o(z,s-'ys)

& [u,z] =[v,z]= [ u x ' , z t I = [ v y ' , z t ] = [ u x ' t , z - ' k - ' v y '

R2(x,y): Rn+l(x,Y):

2 3 x =1 & [x,yl =1 & I x , y l # l , Rn(x,y)

'where x '

, y'

k ] = [ v y ' t , z - l h - ' u x ' h ] =1) 1

are short f o r

r - l x r , s- 1y s .

& tX,Y n 1 2

A s t r a i g h t f o r w a r d a p p l i c a t i o n o f t h e lemmas o f 12 e x p l a i n s t h e meaning of t h e s p r e d i c a t e s f o r t h e groups

Sa, where

t e g e r s l a r g e r than 6, and

s,=~.

CY

ranges o v e r

w, w + l

and a l l p o s i t i v e i n -

163

Symmetric Groups and the Open Sentence Problem

I f the constants

LEMMA 4. t,,,

a

and

b

a r e i n t e r p r e t e d by

and

T

t h e n , i n t h e g r o u p S a + l , t h e e l e m e n t s 5 , 0 and 5

6 < a 5 w,

satisfy the predicate D(x,y)

iff

E = c 9 and

C(X)

iff

E=cq,

E(x,y)

iff

c=< 9

C(x,y,z)

iff

5=cp,

n=cq

5=cP,

9

f o r some qcN w i t h 2 q < a , 9' f o r s o m e qcN w i t h 2 q < a , rl=<

and r l ~ S ~ + f o~r ,some qcN w i t h Zqca, and c = <

and II(x,y,z)

iff

f o r some p , q ~ Nw i t h 2 p , 2 q < a ,

P+9'

and < = < P'9'

f o r some p , q ~ Nw i t h 2 p , 2 q < a .

The next two lemmas deal with the language

of a permutation

IT

o f groups.

The support

consists of a l l points t h a t are moved by i t .

a single o r b i t o f cardinality larger than one. i s a subset of

L

151 on which the action of

A cycle

The o r b i t o f a sub cyc1e;rl coincides with t h a t of

rl

for a t the single point in which the supports o f

rl

and

En-'

,€

5 has of

5

5 except

intersect.

of length one has t r i v i a l action, the length of a non t r i v i a l cycle

In[

A cycle

coincides

with the cardinality of i t s support and with i t s order, the predecessor of which we denote by LEMMA 5.

v(E).

For 6 < a 5 o + l :

(i)

S a I= T ( S )

(ii)

Sa b S ( c , r l )

€,ES is a transposition, i.e.,

iff

iff

=s

B+1'

€, i s c o n j u g a t e t o

f o r some B < a , by a n i n n e r a u t o m o r -

phism i n c a s e B
Sa C I ( € , , r l )

iff

=Sa+l,

( i i i ' ) So k (WX)(WY)-G(X,Y),

iff

t h e s u p p o r t s o f t h e c y c l e s 5 and

iff

rl

i n one p o i n t , (v)

Sa C O(€,,rl)

LEMMA 6.

5= 1

iff

If 6 < a

5

i s a sub c y c l e of t h e c y c l e w + l and c,q,c

E

Sa+l,

€, i s a f i n i t e c y c l e w i t h v(€,)=O,

( I t ) ( 3 g ) (3h) ( S ( t , t ( )

& n=g-'t€,g=h-'n-'h)

then

5.

n intersect

V. HUBER-DYSON

164

4.

COBHAM'S THEOREM Lemma 6 shows how t o define predicates

for ncN, Sc, Sm*, Pr* and O* in Cn, the language of groups so t h a t the axioms of the fragment Ro of arithmetic become true in every i n f i n i t e model of the theory of f i n i t e symmetric groups.

Ro

encodes

the diagram of the arithmetic of the natural numbers under successor, addition, multiplication and the natural ordering, using predicates rather t h a n terms and has the additional axioms

(Vx)

- O*(x,l)

and

(Wx)(Yy)(Vz)(Cn(x)& Sc(x,z) & O*(y,z) => Cn(y)VO*(y,x)), f o r Every f i n i t e subset of

i s undecidable.

Ro

has f i n i t e models, b u t every theory compatible

embeds a copy o f

S

Using the f a c t t h a t every s u f f i c i e n t l y large alternating group

n v i a . t h e diagonal Sn

nating groups where the cycles

<,

with Ro

For a discussion and a proof yielding stronger r e s u l t s the reader

i s referred t o [8].

of

ncN.

+

SnxS,

Lemma 6 can be proved f o r a l t e r -

a r e replaced by products of two d i s j o i n t copies

and we arrive a t

THEOREM 1 (Cobham). Every c l a s s o f g r o u p s t h a t c o n t a i n s i n f i n i t e l y many f i n i t e s y m m e t r i c o r a l t e r n a t i n g g r o u p s h a s a n u n d e c i d a b l e element a r y theory. I t follows t h a t the theories of f i n i t e groups, simple groups, many more.

of monotlithic

of complete groups,

of f i n i t e

and o f periodic groups are a l l undecidable, and so are

Note however t h a t Mal'cev's r e s u l t [4] of the hereditary undecidability

o f the theory of f i n i t e groups of exponent p , f o r prime p > 2 , and of nilpotency

Symmetric Groups and the Open Sentence Problem

165

class 2 i s not covered by Cobham's theorem.

5.

THE UNDECIDABILITY OF Neither of t h e groups

AND

S

Sw.

In

i s a model f o r t h e theory of f i n i t e groups.

S , Sw

f a c t t h e i r t h e o r i e s a r e compatible with t h e f i n i t e l y axiomatizable e s s e n t i a l l y undecidable fragment Q

of a r i t h m e t i c .

Thus t h e u n d e c i d a b i l i t y of t h e i r elementary

theories follow by t h e c l a s s i c a l methods o f [ 7 ] . relativize t o the predicate N

For

S

i t i s most convenient t o

and use Lemma 4 t o obtain an i n t e p r e t a t i o n of

in the i n e s s e n t i a l extension of the theory TS by t h e constants Sw

relativize t o the predicate

Lemma 6.

a

Q

and b.

For

( 3 z ) S ( z , x ) and use t h e predicates introduced in

The c r u c i a l d i f f e r e n c e between i n f i n i t e models of t h e theory of f i n i t e

symnetric groups and

lies i n Lemma 5 ( i i i ) , ( i i i ' ) .

Sw

i n t e r p r e t a t i o n s of the axioms of f i n i t e group theory and

Ro

Q

The conjunction of the

i s of course t h e negation of a theorem of

has t o be used f o r Cobham's proof, but i n

So

every

cycle has a "successor", and a l l "sums" and "products" e x i s t .

THEOREM 2.

The e l e m e n t a r y t h e o r i e s o f b o t h t h e g r o u p o f a l l p e r -

m u t a t i o n s o f f i n i t e s u p p o r t o n an i n f i n i t e s e t a n d i t s e x t e n s i o n b y a f i x p o i n t f r e e cycle are h e r e d i t a r i l y u n d e c i d a b l e .

6.

THE EQUATION PROBLEM FOR SYMMETRIC GROUPS WITH A DISTINGUISHED PAIR OF GENERATORS Lemma 4 shows how t o a s s o c i a t e w i t h every polynomial equation

integral c o e f f i c i e n t s and v a r i a b l e s $ ( a , b ) of t h e language

L'

f r e e v a r i a b l e s x1 ,...,xk,yl

nl

,...,n k

and only i f

S = Su+l.

t h e equation

x1 ,. . . ,xk,y,

,. . . ,yh

an e x i s t e n t i a l formula

of groups enriched by two constants

,...,y h

P(nl

a, b

and w i t h

such t h a t , f o r every choice o f natural numbers

,..., nk,yl ,...y h )

$(nl,...,n+,yl ,...,y h )

P w i t h positive

has a n o n t r i v i a l s o l u t i o n i n

N

if

i s s a t i s f i a b l e i n t h e permutation group

The u n s o l v a b i l i t y of H i l b e r t ' s t e n t h problem, [51, thus e n t a i l s t h e un-

d e c i d a b i l i t y of t h e equation problem f o r

S

in terms of the generators

T

and

B u t a b i t more can be s a i d upon s c r u t i n y of the language introduced i n 53. All

p.

166

V. HUBER-DYSON

terms t h a t occur in I) will be of t o t a l exponent zero in words of the form v(a,b) s a t i s f i a b l e in

S

t h a t in f a c t belong t o Sw,

so t h a t our formula i s

iff

Sw+l k ~ ( n , ( a , b ),...,~ , ( a , b ) q, ( a , b ) f o r some ml, ...,m h

and the solutions are

b

E

N.

,..., rn+(a,b))

B u t then, there i s an integer

q

(d

such t h a t

Sq+l k I ) ( q ( a , c ),. . . , s ( a , c ) , q ( a , c ) ,. . . ,rn+(a,c)).

q

(9)

d i l l be a p a r t i a l recursive function of the n ' s with the property t h a t i f

(w)

holds then the sentence on the r i g h t will be a logical consequence of the f i r s t q

if

defining relations of

(9)

r > q,

[a,b] # 1 .

and the inequation

So+l

holds f o r a formula of the form indicated, then so will including

w.

On the other hand, (r)

for a l l

Noting the recursiveness of the presentations involved and

modifying the notation of I1 by writing

W', 3'

where the language

i s con-

L'

cerned and attaching a subscript 0 i f we mean r e s t r i c t i o n t o formulas in which only terms of t o t a l exponent 0 i n b

occur we find t h a t

5 30' = 5m3O' = 3$

m

= 3;

s

and t h a t a l l these s e t s are recursively enumerable b u t n o t recursive.

I t follows

t h a t the s e t s 53', 3'5, and 3's a r e recursively enumerable b u t n o t recursive and

t h a t none of the s e t s in the following sequence i s recursive S3' 3 Sm3'

2

3 ' S m 3 3'S3 3 '

V ' g c V'SmC W'S

To see t h a t the inclusions are proper note t h a t b

b9 = 1

in only f i n i t e l y many

Sn's,

i s a commutator in i n f i n i t e l y many b u t n o t in almost a l l of them, for fixed

n , m , c(n,m,y) b

.

s

i s s a t i s f i a b l e i n almost a l l symmetric groups b u t n o t in a l l , and

i s conjugate t o i t s inverse in a l l f i n i t e symmetric groups b u t n o t in

S.

THEOREM 3. The s e t s o f q u a n t i f i e r f r e e f o r m u l a s o f t h e l a n g u a g e o f

g r o u p t h e o r y e n r i c h e d by t w o c o n s t a n t s - - i n t e r p r e t e d by a t r a n s p o s i t i o n and a maximal c y c l e

--

t h a t a r e s a t i s f i a b l e i n some, i n a r b i t r a r -

i l y l a r g e o r i n almost a l l f i n i t e symmetric groups a r e u n d e c i d a b l e , and so i s t h e e q u a t i o n p r o b l e m f o r t h e g r o u p S,

S.

The o p e n t h e o r i e s o f

o f a l m o s t a l l f i n i t e s y m m e t r i c g r o u p s and o f a l l f i n i t e s y m m e t r i c

Symmetric Groups and the Open Sentence Problem

167

groups in the extended language are not recursive. The p r e d i c a t e

E

o f 53 a l l o w s a r e l a t i v e i n t e r p r e t a t i o n o f t h e t h e o r i e s o f

a l l f i n i t e symmetric groups, o f a l l f i n i t e a l t e r n a t i n g groups and o f almost a l l f i n i t e symmetric groups i n t h e t h e o r y o f t h e group

A p p l i c a t i o n s w i l l be i n v e s -

S.

t i g a t e d elsewhere,

7.

REDUCTION TO THE LANGUAGE OF GROUP THEORY I t should be p o s s i b l e t o sharpen theorems 1, 2 and 3, b u t a t t h i s p o i n t t h e

f o l l o w i n g remarks w i l l have t o s u f f i c e .

U s i n g Lemna 5 ( i i i ) and t h e a s s o c i a t i o n o f

86 between p o l y n o m i a l e q u a t i o n s and f o r m u l a s b u i l t up from t h e p r e d i c a t e s and terms

o f 83 one f i n d s t h a t t h e s a t i s f i a b i l i t y i n some f i n i t e symmetric group of t h e 3W3formula

(3a)(3b)(G(a,b)

,... ,n+.y ,,... ,yh))

i s equivalent t o t h & s o l v a b i l i t y

& $J(~I~

of t h e corresponding d i o p h a n t i n e problem.

Moreover, due t o t h e form of

$J.

the

formula i s s a t i s f i a b l e i n some f i n i t e symmetric group i f and o n l y i f i t i s so i n almost a l l o f them and a l s o i n t h e group

THEOREM 4.

S.

The W3W-theories of finite symmetric groups, of almost

all finite symmetric groups and o f the group

S

of permutations gen-

erated by a transposition and a fixpoint free cycle on an infinite set are not axiomatizable. o f 83 and t h e f a c t t h a t 9 t h e u n i v e r s a l t h e o r y o f a l l f i n i t e groups c o i n c i d e s w i t h t h a t o f 5, [3], one o b t a i n s

W i t h t h e p r e s e n t a t i o n s o f Lemma 3, t h e p r e d i c a t e s

R

THEOREM 5. For any quantifier f r e e formula

H

groups let

of the language of

abbreviate the W3-sentence 9 (Vx)(Vy)(Rq(x,y) =' ( 3 2 1 ) ...(3zk)H(x,y,zl,...,zk))'

(i)

H

For each positive integers

q , either

symmetric group of degree greater than of group theory for all

q,

H

9

fails in some finite

or e l s e

Hr

is a theorem

r 2 q.

(ii) The set of quantifier free formulas

H

for which

H is, for 4

V. HUBER-DYSON

168

q, a theorem o f finite group theory is recursively enumerable

some

but not recursive. Finally, the presentation f o r

S

on

exponent z e r o i s an elementary p r o p e r t y o f troducing the predicate (WZ)(X

2

and

T

and t h e f a c t t h a t b e i n g o f

p,

as an element o f

p

S

justifies in-

A(x,y):

2 2 1 = l & [ x ,y13=1& [x ,y1 f l & y f z & ( [ z ,y1 = l & z = l = > z = l ) & ( [z, y1=1=>z=yz=y- v [ z , x l 2 = 1 ) f .

Any two elements o f a group isomorphic t o

S

and

G

that satisfy

A

in

G

w i l l generate a subgroup

A ( T , ~ ) holds i n the f r e e product o f

S

w i t h any group.

From 56 and Theorem 2 o f [3] one o b t a i n s a theorem t h a t , amusingly, has t h e undec i d a b i 1 it y o f Wsgrouptheory as a c o r o l l a r y . THEOREM 6.

by the axiom arithmetic.

The extension (3x)(3y)A(x,y)

T

of the elementary theory of g r o u p s

is compatible with the fragment

Q

of

Its universal theory coincides with that of all groups

and is undecidable while the existential closures of exactly those quantifier f r e e formulas that are finitely satisfiable are theorems of

T.

The set of quantifier free formulas

(Wx) (Wy) (A(x,y) = > (3 zl).

,

H

for which

. (3zk)H(x,y,zl,. . . ,zk))

grouptheory is not recursive.

is a theorem o f

Symmetric Groups and the Open Sentence Problem

169

REFERENCES

A . Cobham, Undecidability in group theory, AMS Notices, vol 9 (1962), 406. V. Huber-Dyson, The word problem and r e s i d u a l l y f i n i t e groups, AMS Notices, vol 11 (1964), 743. V. Huber-Dyson, A reduction of the open sentence problem f o r f i n i t e groups, t o appear in t h e B u l l e t i n of t h e LMS, vol. 13.

A. I . Mal'cev, The u n d e c i d a b i l i t y of t h e elementary theory of f i n i t e groups,

Dokl. _ _ Akad. Nauk, SSSR 138 (1961), 771-774.

Y.

v.

Dokl. Akad. Matiyasevich, Enumerable s e t s a r e Diophantine, - Mauk. -

SSSR 191 (1970), 279-282.

C . F. M i l l e r 111, Some connections between H i l b e r t ' s 10th problem and t h e theory of groups, i n m p r o b l e m s , ed. W . W . Boone, F. 8. Cannonito and R . C . Lyndon, North Holland Co. 1973. A . T a r s k i , A . Mostowski and R. M . Robinson, Undecidable Theories, North Holland Co. 1953. R . L . Vauaht. On a theorem o f Cobham concernina undecidable t h e o r k s . i n t h e Philosophy of Science, ed. E . Nagel, P . Suppes and A. T a r s k i , Stanford Univ. Press 1962.

w ,Methodology and