A K A D E M I E D E R W I S S E N S C H A F T E N D E R D D R

I N S 1 ' I T U T F U R M A T H E M A T I K

Jakov Ju. Gol'fand, Nina L. Najmark, and Reinhard Poschel

The structure of S-rings over Z2,,,

B e r l i n

AKADEMIE DER WISSENSCHAFTEN DER DDR

INSTITUT FUR MATHEMATIK

Jakov Ju. Gol'fand, Nina L. Najmark, and Reinhard Poschel

The structure of .S-rings over Z 2m

(Communicated by A. Baudisch)

Berlin 1985

Keywords

S-ring (Schur r i n g )

Permutat i on group

AMS Subject c l a s s i f i c a t i o n (1980)

20C05, 05C25, 20B99

Received November 6 th , 1984

Summary; The f u l l d e s c r i p t i o n o f t he s t r u c t u r e o f a l l S - r i ngs

(Schur- r ings) over Zn, the c y c l i c group o f o rde r n, p rov ides

much i n f o r m a t i o n on both, permuta t ion groups c o n t a i n i n g a re -

g u l a r r e p r e s e n t a t i o n o f Zn and graphs w i t h auch automorphism

groups. I n the present paper, t he f u l l d e s c r i p t i o n o f t he

s t r u c t u r e o f a l l S- r ings over Z (m a r b i t r a r y ) i s given, 2m

T h i s r e s u l t answers the l a s t open case f o r t h e d e s c r i p t i o n o f

S- r ings over c y c l i c groups o f pr ime power order , which now

can be considered as known,

Zusammenfassung. D i e v o l l s t B n d i g e Beschreibung a l l e r S-Ringe

(Schurschen Ringe) uber de r zyk l i schen Gruppe Zn de r Ordnung

n l i e f e r t v i e l e wesent l iche In fo rma t ionen sowohl uber Permu-

tat ionsgruppen, d i e e i n e r e g u l l r e D a r s t e l l u n g von Zn en tha l -

ten, a l s auch Gber Graphen m i t d e r a r t i g e n Automorphismengrup-

pen. I n d e r vor l iegenden A r b e i t w i r d d i e S t r u k t u r a l l e r S-

-Ringe eber Z f u r b e l i e b i g e s m beschrieben, Damit w i r d zu- 2m

g l e i c h das Problem zum AbschluB gebracht, d i e S t r u k t u r a l l e r

S-Ringe uber zyk l i schen Gruppen von b e l i e b i g e r Primzahlpo-

tenzordnung zu beschreiben.

P e s ~ ~ e . 3 ~ 1 m a e p e r n g ~ ~ ~ Bcex S-rcoxes (wo~req Dypa) ~ H K X H -

qec#oH rpynnoH Zn nopnma n A ~ G T 6 o r a ~ y ~ a~@op~xaanD Aan ona-

CaHHR KBX FpyIXIl IIOACTaHOBOK , COAepXaaMX PeryJIEpHOe ripe ACTaB-

nemrtre z Tax H rpa@oB c T ~ K B M R rpy-mam a B ~ o ~ o p @ x a ~ o B . B n'

mrac~o~rsef p a d o ~ e onacsvro crpoerrne ~ c e x S-xonen HaA z npa 2m

npoKsBonsHoM m . T ~ M c a m saBepmeHo pemeaxe saAam onncasrnR

CONTENTS

In t roduct ion ........................................... 3

9 1 Pre l iminar ies ....................................... 4

92 The computer algorithm f o r the construction

of a l l S-rings ...................................... 7

$33 The Theorem ......................................... 10

54 Proof (Par t I ) ...................................... 16

55 Proof (Par t 11) ..................................... 22

Referenbea .............................................. 28

INTRODUCTION

The method o f so-ca l led S- r ings (Schur- r ings) i s well-known

w i t h i n the theo ry o f permuta t ion groups. I t goes back t o I.

Schur [10] and has been developed f u r t h e r b y R. Kochendi j r f fer

[6] and H. Wie landt fi3J ( f o r b a s i c r e s u l t s we r e f e r t o 172, [10],[13],[14J,Jll],J9]). I n the l a s t years, an i n t e r e s t i n g new

a p p l i c a t i o n o f t h i s method was found: the a p p l i c a t i o n t o the

isomorphism problem f o r some c lasses o f Cayley graphs ( c f . J41,

153, f9_J). Hereby the c r u c i a l p o i n t t u r n s ou t t o be the desc r ip -

t i o n o f t he l a t t i c e o f S-r ings over a r e g u l a r r e p r e s e n t a t i o n o f

the group under cons ide ra t i on . The f u l l d e s c r i p t i o n o f a l l S-

r i n g s over the r e g u l a r rep resen ta t i on o f a c y c l i c group o f o rde r

pm (where p i s an odd pr ime) was g i ven i n p]. The case p = 2

remained open.

I n the present paper the d e s c r i p t i o n o f a l l S- r ings over Z

i s given. A f i r s t approach f o r s o l v i n g t h i s problem was done 2m

b y u s i n g a computer (o f t ype ES 1020). W i th t h i s computer pro-

gram i t was p o s s i b l e t o determine a l l S-r ings over z2"

f o r

1 C m f 6 . Examining the r e s u l t s ob ta ined i n t h i s way, a genera l

con jec tu re on the s t r u c t u r e o f S-r ings over z2"

was found and,

f i n a l l y , proved. Some o f these r e s u l t s ( w i t h o u t p r o o f ) has been

announced i n D] and, i n another terminology, i n [15J.

I n the present paper, t he theorem on the s t r u c t u r e o f S- r ings

over Z 2m

w i l l be g iven i n 53, t he p roo f i s g i ven i n 554,5.

A f t e r some d e f i n i t i o n s i n § l , a s h o r t d e s c r i p t i o n o f the compu-

t e r a l g o r i t h m a n d t h e l i s t o f S- r ings over Z i s g i ven i n 52. 8

ACI~NO~VLEDGEPIEN~S. The au thors wish t o express t h e i r thanks t o

M.H. l t l i n f o r many u s e f u l h i n t s and s t i m u l a t i n g remarks and f o r

h i s cont inuous i n t e r e s t t o t h i s paper. Our thanks a re a l s o due

t o I.E. ~a rnecka ja fo r u s e f u l cooperat ion.

Qi PRELIMINARIES

I n t h i s s e c t i o n we s h a l l g i v e t h e most i m p o r t a n t f a c t s and de-

f i n i t i o n s concern ing S - r i n g s ove r c y c l i c groups. F u r t h e r i n f o r -

ma t i ons (sometimes used i n t h i s paper) can be found e.g. i n [13J

[91.f12]

L e t H be some group and Q(H) i t s groue .-- r i n g ( o v e r t h e f i e l d

o f r a t i o n a l numbers Q) . The o p e r a t i o n o f m u l t i p l i c a t i o n i n t h i s

group r i n g w i l l be denoted b y r . The e lements o f Q ( t l ) . i .e .

t h e f o r m a l sums (chcQ), a r e c a l l e d g u a n t i t i e s . _--_---_- I f

K P H . t hen g k i s c a l l e d --. simele -- g u a n t i t y --I--- and w i l l be denoted

b y - K . A s u b r i n g R o f Q(H) i s c a l l e d an S - r i n g ----- o v e r H, i f

a) R i s a submodule o f Q(H) and has a b a s i s o f s imp le

q u a n t i t i e s To. T1. .... T, ( b a s i c ----m q u a n t i t i e s ) , I-_---m-- where - - - b ) To = {e) ( e t h e u n i t e lement o f H ) ,

S C ) Ti = H i s a p a r f i t i o n o f H. i .e. TinT = p ( i # j ) .

i = o 3

I n t h e f o l l o w i n g , we s h a l l c o n s i d e r S - r i ngs o v e r t h e c y c l i c

group <z,;+>, n ) 2. Zn w i l l be w r i t t e n a d d i t i v e l y and t o a v o i d

c o n f u s i o n s we s h a l l l i s t he re e x p l i c i t e l y a l l o p e r a t i o n s and no-

t a t i o n s i n S - r i n g s i n t h a t form, which w i l l used th roughou t t h i s

paper: L e t t,keZ,. c r z ( i n t e g e r ) T ,Ks Zn.

~ d d i t i o n i n t h e groue r i n g / f o r m a l suml: S i g n t, ( o r +,E -------m-------- --- ---- - ---------- - if i t i s c l e a r f rom t h e c o n t e x t t h a t + i s t h e S - r i n g a d d i t i o n ) ;

Example: - T t m K (= g h o r I+K he K

Sca la r m u l t i e l i c a t i o n i n the groue r i n g : --------.--m I--------------- --- W--- -

i n p a r t i c u l a r , r l

= G c k ( c w r i t t e n on the l e f t s i d e o f 5, 2).

A d d i t i o n and s u b t r a c t i o n modulo n: S ign + , b ------------m---------------------

Examples: t+z, t -z, T + z:={t+z I t 6 ~ 3 ,

T a K := T + K := { t + k t r T , k e ~ ) . M u l t i e l i c a t i o n - - m - - ---.---------W---- modulo n: t k (sometimes a l s o t e k ) ,

i Examples: A 2 (= ~2' mod n),

Tek := ( t k I t r ~ ) , ~ * l < : = { t k ( t r T , krK],

p q = > := ( t q ) f o r q&Z w i t h g.c.d.(q,n)=l

( q w r i t t e n on the r i g h t s ide o f L. T) . M u l t i e l i c a t i o n i n the S-r ing: S ign n ----- --W------------------ - Example: T s K = f ( t + k ) - - .

t e krlc

We note the f o l l o w i n g r e s u l t o f Schur/wielandt ( c f .D, 8.3.131)

which w i l l o f t e n be used ( w i t h o u t re fe rence) i n t h i s paper:

L e t I TIR be a s imple q u a n t i t y o f an S - r i ng R over In ( o r

over an a r b i t r a r y a b e l i a n group). Then I Taq a l s o belongs t o R

p rov ided t h a t g.c.d.(q,n)=l. I n p a r t i c u l a r , i f I T and - T ' are

b a s i c q u a n t i t i e s w i t h 1&T and q r T e , then T @ = T ~ ~ = X I . L e t D={do.dl,-..d k - l I be the se t o f a l l d i v i s o r s (#n) o f n

and 1 . For K 6 I,, dtDn d e f i n e

Moreover, l e t

be the b a s i c q u a n t i t y o f R which con ta ins x(Z, ( i .e. xaT ). ( X )

The b i n a r y r e l a t i o n

e ( ~ ) = { ( d , d @ ) t ~ ~ a o ~ I T(d) /d ' # p 3

i s an equiva lence r e l a t i o n on Dn and i t i s c a l l e d the b a s i c ----- eguiva lence - --------- o r the eguiva lence o f the S - r i ng R . An equiva lence - -----m--- -0 --- ----m

8 i s c a l l e d admiss ib le ---------- i f 8 = 8 ( R ) f o r some approp r ia te S- r ing R.

L e t P ( n ) be the ( m u l t i p l i c a t i v e ) prime res idue c l a s s group

o f a l l numbers ( o f Zn) r e l a t i v e l y pr ime t o n. Then

G(d) : = { x r ~ ( n ) I T ( ~ ) . x = T ( ~ ) ] (de~,)

t u r n s ou t t o be a subgroup o f P ( n ) . The sequence

i s c a l l e d the S-system o f -- --- the S- r ing ---m- cf. f83).

We are going t o determine a l l subgroups o f ~ ( 2 ' ) . m > l.

(Throughout the paper, working w i t h elements o f Z 2m , a11 pro-

duc ts and sums a re taken modulo 2m). Consider the f o l l o w i n g sub-

s e t s o f ZZm, which a re a l s o subsets o f ~ ( 2 ~ ) :

These subsets a re subgroups o f ~ ( 2 ~ ) ( b e c a u s e they a re c losed

under m u l t i p l i c a t i o n mod 2m). Moreover, they a re a l l subgroups

o f ~ ( 2 ~ ) . T h i s can be seen e i t h e r d i r e c t l y (because the s e t

4-1.5) generates ~ ( 2 ~ ) = { 5 ' 1 0 1 r < 2m-2)~{ -5 r 1 0 g r < 2 m-2)

( c f . [l)) and a l l subgroups must have genera t ing elements o f the

- { - 1 . 5 ~ ~ > , form X3i+l- 2 1

~ ~ ~ + ~ ~ ( 5 )(o 5 i f m-2) ~ ~ ~ = < - 5

( 1 C i f m - 2 ) ) ,

o r by count ing the subgroups o f ~ ( 2 ~ ) (us ing the well-known

f a c t [l], t h a t ~ ( 2 ~ ) i s i somorph ic t o t h e d i r e c t p roduc t

t h e r e f o r e e v e r y subgroup i s a s u b d i r e c t p roduc t *2 " z2m-2

o f a subgroup o f Z2 and a subgroup o f Z 2m-2 ; o b v i o u s l y t h e r e

i s one s u b d i r e c t p roduc t o f Z1 and Z2kdZ2m (namely ZlxZ 2k)

and, b y a c l a s s i c a l theorem ( I< le in /F r i c l te ) , t h e r e a r e two

s u b d i r e c t p r o d u c t s o f Z2 and Z ( f o r k ~ 1 , f o r k=O t h e r e i s 2k

o n l y one) 3 t h u s we have 3(m-2)+2 = 3m-4 subgroups o f ~ ( 2 ~ )

(one o f o r d e r 2m-1B one o f o r d e r 1 ( t r i v i a l ) and f o r a l l

m - i l e i e m t h r e e o f o r d e r 2 ) ) . Cons ide r i ng t h e above groups we no te , t h a t X3i+2 (0s ism-2)

i s t h e s t a b i l i z e r o f t h e element 2 m-i-zeZ

f .e.,

m-i-2 2m ' X3i+2={~ ' ~ ( 2 ~ ) I 2m-i-2.x=2 , and Xi i s t h e s t a b i l i z e r

o f 2m-1 . Moreover we have X3i+2 C X3j, X3i+2 C X3j+1 and

' 3 i - t ~ 1 '3j+2 f o r j s i .

5 2 THE COMPUTER ALGORITHPI FOR THE CONSTRUCTION OF - ALL S-RINGS

I n p r i n c i p l e , t h e problem o f de te rm in ing a l l S - r i ngs o v e r

Zn can be v iewed as a s p e c i a l case o f d e t e r m i n i n g a l l sub-

V - r i ngs ( c e l l u l a r subrings(i16JB coheren t c o n f i g u r a t i o n s (kl ig-

man), c f . D , p.1851) o f t h e V - r i ng o f t h e g i v e n pe rmu ta t i on

group ( h e r e a r e g u l a r r e p r e s e n t a t i o n o f Z ) The g e n e r a l i d e a

f o r an a l g o r i t h m which l i s t s a l l such c e l l u l a r s u b r i n g s was

g i v e n i n [2J. On t h i s bacl<ground, a b a c k t r a c k i n g a l g o r i t h m

was worked o u t b y V.A. ~ a i E e n l c o d e t e r m i n i n g S - r i ngs w i t h some

t e n g e n e r a t i n g q u a n t i t i e s . I n t h e s p e c i a l case o f S - r ings ,

t h e a l g o r i t h m i s based on t h e n o t i o n o f t h e S-system which

c o r r e s p o n d s t o e a c h S - r i n g a n d w h i c h w a s i n t r o d u c e d i n DJ. T h e a l g o r i t h m c o n s i s t s o f t h e f o l l o w i n g p a r t s ( c f . a l s o B],

J41): a ) D e t e r m i n e a l l s u b g r o u p s o f l P ( n ) ,

b ) D e t e r m i n e a l l a d m i s s i b l e e q u i v a l e n c e r e l a t i o n s o n D,,

c ) F o r e v e r y e q u i v a l e n c e , d e t e r m i n e a l l S - s y s t e m s

" c o m p a t i b l e w w i t h t h i s e q u i v a l e n c e ,

d ) F o r e v e r y p o s s i b l e S - s y s t e m , d e t e r m i n e a l l " c a n d i -

d a t e s " f o r S - r i n g s ( i . e . , c o n s t r u c t a l l b a s i c q u a n -

' t i t i e s o f t h e c a n d i d a t e ) .

e ) C h e c k w h i c h o f t h e c a n d i d a t e s a re S - r i n g s .

S u c h a n a l g o r i t h m a n d a c o m p u t e r p r o g r a m w e r e w o r k e d o u t i n

d e t a i l f o r t h e c a s e n=Zm, T h e FORTRAN-program c o n s i s t e d o f

t h r e e p a r t s :

T h e f i r s t p a r t g a v e t h e l ist o f a l l e l e m e n t s o f a l l s u b g r o u p s

o f P ( n ) i n t h e e n u m e r a t i o n f i x e d a b o v e i n §l. I n t h e s e c o n d

1 p a r t , a l l a d m i s s i b l e e q u i v a l e n c e s 0 o n D 1 f 2 O . 2 , ,.,2 m - 1 )

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

e q u i v a l e n c e a l l p o s s i b l e S - s y s t e m s h a v e b e e n c o n s t r u c t e d ,

i .e . s e q u e n c e s o f i n d i c e s o f s u b g r o u p s ( t a k i n g i n t o c o n s i d e -

r a t i o n t h a t t o t h e d i v i s o r d = 2 m - i c a n c o r r e s p o n d o n l y s u b -

g r o u p s w h i c h c o n t a i n i t s s t a b i l i z e r X3i - . W i t h t h e S - s y s -

tems (Go,G1, "., G m - l ; Q ) a l l " c a n d i d a t e s " f o r S - r i n g s R a r e

g i v e n , u s i n g t h e f a c t , t h a t f o r e v e r y e q u i v a l e n c e c l a s s I<

o f 8 t h e c o r r e s p o n d i n g b a s i c q u a n t i t y h a s t o b e t h e u n i o n o f

o r b i t s o f t h e g r o u p Gi a c t i n g o n t h e se t Z /zi b y m u l t i p l i - 2"

c a t i o n mod 2m. A f t e r t h i s , t h e p r o d u c t o f e v e r y two b a s i c

q u a n t i t i e s h a s b e e n c o m p u t e d a n d i t w a s c h e c k e d w h e t h e r t h e s e

p r o d u c t s a r e l i n e a r c o m b i n a t i o n s o f b a s i c q u a n t i t i e s . A l l

c a n d i d a t e s f u l f i l l i n g t h i s p r o p e r t y g i v e t h e f u l l l ist o f a l l

S-r ings. The computat ions f o r n=8,16,32,64, resp., has been

performed on a computer o f type EC 1020, t i d y the re were

found 10 (7 ) , 37(28), 151(120), 657(538) S-r ings, resp.,

where t h e numbers i n paran thes is g i v e the number o f S- r ings

R w i t h t r i v i a l equiva lence ~ ( ~ ) = { ( 2 ~ j , ( 2 ~ f , ,. , (2m'1)),

As an example we g i v e the l i s t o f a l l S- r ings over zn 3 f o r n-8x2 (m=3)( for m=4, m 4 see e.g. D]) t

+) i n d i c a t i n g (Go,G1,G2)=(Xi .X #Xio) f o r the subgroups o f 2 id

the corresponding S-system.

. b a i i c q u a n t i t i e s o f the

S- r ing

0, 1,3,5,7, Q, 2 9

- 0, 1,3,5,7, - 2, - 6, - 4

- A A A - 0, 1 5 , 3 7 , 2 6, 4

g, 0 A* 1 5 L' 3 7 - 2 , 6 , 4 g g

- 0 8 Q 8 5 , 7 8 2 , 6 8 5 - A A A - 0, 1 7, 3 5, 2 6, 4

- O, A 8 21 58 L8 28 g8 5

0, 1,3,5,7,2,6, g 4

g 0, 1,3,5,7, 2,G84

g 0, 1,2,3,4,5,6,7 .

. urn- admiss ib le ( i z , l,, iO! b e r equiva lence 1 8

Remark: Computer a lgo r i t hms which work w i t h genera l V-r ings

( i n p a r t i c u l a r w i t h S-r ings) can be found a l s o i n /16],[17]

( b a s i c r e s u l t s and e p p l i c a t i o n e ) .

2

3

4

5

6

7

8

9

10

II

n

I*

n

n

II

{b0,21),i22JJ

1 2 {12°},{2 82 f i

1 2 {{2082 ,2 J {

(1,2a1)

(2,181)

(2,281)

(381,1)

(4,181)

(58281)

(18181)

(181.1)

(1,181) -

S3 THE THEOREM

The f u l l desc r i p t i on of a l l S-rings over Z ( f o r a r b i t r a - 2m

r y m) has been found on the background o f the computer re-

s u l t e ( c f , §2), We are going t o formulate the f i n a l r e s u l t ,

bu t before doing that , we need some more no t ions and nota t ions.

We s h a l l eee t h a t an S-r ing R over Z w i l l e x i s t i f and 2f"

o n l y i f i t s equivalence Q(R) cone is ts of equivalence classes

o f the form

+ pi+J) f o r some i , 3 ~Io, 1 , ... , m - 1 1 I 2 . 2 #..*#

only, For the moment, equivalences o f t h i s form we w i l l c a l l

eguivalencee W --------- - of mll&clg&b&) fg~"because they are e x a c t l y the

admissible equivalences) . E ,g., f o r m-3, a l l equivalences of

admissible form are the f o l l o n i n g : {2l), 1 2 ~ ~ 3 , 1 2 {{20,2~1 ,{22fi {fzO) , { ~ ~ , 2 ~ ) 1 , fl20,2 ,2 f3. ~t i. easy t o see,

0 1 ''l - 2*" equivalences o f t h a t there are Cmol + Cmol + m, + Cmrl

admissible form f o r S-rings over Z 2'"

We s h a l l descr ibe the set o f a l l S-systems (o f , 51) w i t h

a g iven equivalence Q of admissible form by a labe led rooted

t r e e &Q) :

m - 1 For the t r i v i a l equivalence h, -{i2°1 , 2 j) t h i s t r e e d(€,) hae m layers. The p o i n t s of the t r e e are

labe led w i t h a number j o f the subgroups X o f ~ ( 2 ' ) ( c f . 91). j

Every path from the r o o t t o the l ea fes w i l l correspond an - C - . - - - - - 7- 7

S-system i n the f o l l ow ing way; I f the path passes through the * C I - m -

v e r t i c e s w i t h numbers ( l abe l s ) 1-io,il,...,imol, then

i s the corresponding S-system.

Descr ip t ion o f the t r e e d ( ~ , ) t The roo t o f 4~,) i s labe led

w i t h 1, the next l aye r has two p o i n t r w i t h l a b e l s 1 and 2,

The successors f o r the next l a y e r s o f the t r e e are def ined by

the f o l l o w i n g r u l e s :

(a) Successors o f p o i n t s w i t h l e b e l s 31+2 (0 J 16 m-2)

are the p o i n t s w i t h l e b e l s 1 end 3 j+2 ( 0 6 j i l+l).

(b) Successors o f po in t s w i t h l a b e l s 31+2 o r 31+3

( 0 5 1 $ m-2) are the po in t s w i t h l a b e l s 2 end

33+1 (0 6 j f i+l) and 3k ( 1 5 k 5 1+1).

(The nth l a y e r i s the l a a t one).

Descr ip t ion o f the t r ees 4 8 ) The t r e e d(8) f o r S-systems

over Z w i t h n o n - t r i v i a l equivelence 8 can be obtained by 2m

'gluelng' together some t rees 4 ( d k ) (k < m ) described above.

More exact ly , l e t 8 be an equivalence of admissible form and

l e t

be the equivalence classee o f 8 w i t h more than one element

(og11*i1+j1<l,~i ,+j2< " * < s < i * + j s m - l , 821). Now.

t o every equivalence c l ass of 8 corresponds a l aye r o f the

t r e e 4 ( 8 ) (which, i n turn, descr ibes the S-systems as above)

m l the roo t corresponds t o [2 - l@, the l a s t l a y e r corresponds

t o [201G. I f the equivalence c lass containes more than one

element, then the corresponding l a y e r degenerates t o one

p o i n t w i t h l a b e l 1, which i s connected w i t h e l l po in t s o f the

preceeding and succeeding layer . F i n a l l y , the t rees 4(ck) ( k number o f one-element equivalence c lasses between the

oonsidered more-element olasses) are f i l l e d i n t o the i n t e r -

v a l l s between two degenerate layere , see F ig . 1.

~ i g . i r The t r e e 4 8 ) ( t h e degenerate l a y e r s a re marked with rr )

l a y e r s

1 ( roo t )

correa onding equiva ! ence c lass

Ae examples, I n Fig. 2. Fig. 3 are g iven the t r e e 4(0) corre-

2 3 spondlng t o the equivalence 842O). {2'j, I 2 .2 3, (241,f25J .b6# 7 f o r n.2 , i.e. m=7, end a l l t reee corresponding t o ell equi-

3 valences of edmissible form f o r n-2 (m.3). The t r ees ere

drawn here h o r i z o n t a l l y , i.e., the r o o t i e on the r i g h t , the

l e a f e s on the l e f t s ide1 the corresponding equivalence c las - . ses are i nd i ca ted below the layers.

~ i g . 2: The t r e e 4 0 )

Fig. 3: The t rees 4 8 ) . B admiseible equivalence f o r m.3

U

For /1(8)(as i n F ig. l ) , every path from the r o o t through

p o i n t s l=lr,l, lr, 2n..., 1, (rnnumber o f equivalence c lasses

of 8) corresponds the S-system

(where 8 has the form as above, Fig, l; note t h a t then e,g.

G = G i+l~w.=G by d e f i n i t i o n o f G (221) ( 2 1 1 ( 2 i l + J l ) ( d ) ) '

Theorem A. For every S- r ing R over Z2., the corresponding

S-system can be obtained from the t r ee *@(R)) i n the above

described #ey. Moreover, there i e a l-l-correspondence - bet-

ween S-r ings over 2 and S-systems obtained as ebove, I.e., - 2'" f o r every S- r ing R, the corresponding S-system can be des- v --v

c r i bed .I a p a t h -v- i n the t r ee ~ ( B ( R ) ) , -* and conversely,

every such path (S-system) crorresponda a unique S-r ing R such - t h a t d i f f e r e n t S-systems correspond non-isomorphic S-rings. - Remark, The desc r i p t i on o f S-rings v i a t h e i r S-systems by

t r e s s 4(8) i s ve ry u s e f u l f o r app l i ca t ions ( i n p a r t i c u l a r f o r

searching a lgor i thms). A more compressed form o f the Theorem

looks as f o l l ows (here the r u l e s (a) and (b) are expressed

e x p l i c i t e l y ) :

Theorem A' [ c f .[3, 2.32). There i s a l - l -connect ion between

S-r ings over Z and S-syatems. A eyeten (Go+G1,,,G,-lrO) 2" -

i s an S-system ( o f an S-ring) i f and 'only i f the f o l l o w i n g -- cond i t ions ere s a t i s f i e d ~

PI 8 -- i s an equivalence - o f admiesible -- form on D m (i,e,, 2

there i s no d i s t i n c t i o n between admissible equivalen- --- ces and equivalence8 of admissible form). --

l1 +j (11) If12 2 ' l i e -- an equivalence -- c l a s s w i t h jl&l

then G = ,. = G - and I1 il+jl= Qil-lm -

Gl m 2 ~ ( ~ 1 . ~ 2 ) (if il=O il=l then de le te the condi- 1

--v-

t i o n s where negat ive i n d i c e s appear). v-

f o r some ( i v ) Let 1 t j L m - 2 . If 0jmX31+l Gj'X3i+3

0 j l j m - 2 - then

c j -1 E { x ~ ) u { x ~ ~ + ~ I 0 C S i + 1 ) v f ~ 3 r I S r 6 "1)

- If O j m X 3 i + 2 - then

=j-1 e I x , l ~ i x ~ . + ~ I Q S. f & + l i ( f o r a g iven S-r ing o r S-system, resp., the concrete s t ruc -

t u r e o f the corresponding S-system o r S-ring, resp., can be

found i n S 1 o r 5 5 , resp.)

The proof o f the theorem i s given i n two p a r t s i n the next

two sect ions t

Par t I (necoessi ty) : We show t h a t every S-system o f an S-ring

over Z muet be o f the form described above, 2"'

Par t 11: We show t h a t f o r every system s a t i s f y i n g ( 1 ) - ( i v )

there e x i s t s an S-ring, the S-sytem o f which equals the g iven system,

Remark. From t h e e t r u c t u r e o f t h e S-systems e a s i l y f o l l o w s ,

t ha t , f o r d i f f e r e n t S-systems, t h e correeponding S-r ings a r e

n o t o n l y d i f f e r e n t , b u t a l s o non-isomorphic.

PROOF (PART I )

s t e p I t A t f i r s t , we a r e go ing t o determine t h e admiss ib le

equiva lence r e l a t i o n s (what proves Theorem A e ( i ) ) . L e t 13.2'

and l e t d be a d i v i s o r o f n (d+n). Then

i s c a l l e d the --W-- t r a c e o f t h e bas i c q u a n t i t y T (d ) *

et R be some S-ring, I t can be eeen w i t h o u t d i f f i c u l t i e s

t h a t t h e s e t o f a l l t r a c e s o f b a ~ i c q u a n t i t i e s o f R generetes

an S - r i ng R' w i t h the seme equiva lence as R: @ ( R ) = @(R0).

Therefore, i n o r d e r t o g e t a l l admiss ib le equivalences, i t i s

s u f f i c i e n t t o s tudy a l l S-r ings o f t race@ on ly . We no te t h a t

0 R i o E R2 t o r R ~ ~ R ~ .

BOCBUBB e v e r y S- r ing over Zn i s conta ined i n Q(Zn), every S-

r i n g o f t r a c e s i s a sub r ing o f t h e S- r ing ~ (2 , )~ . NOW, Q(z,)'

i s t h e maximal S-r ing w i t h the seme equiva lence as Q(2,). T h i e

i m p l i e s t h a t ~(1,)' =<g. P ( ~ ~ L , ~(2')*2,,.., ~ ( 2 ' ) . 2 " - ~ ) i s

t h e t r a n s i t i v i t y module ( c f . e.g. [13, p . 5 g ) o f t h e m-fold

wreath produc t S2 w r S2 w r , w r S2 o f t he symmetric group

s2 o f o r d e r 2. Moreover, ~ ( 2 , ) ~ i s isomorphic t o the V-r ing

( c e n t r a l i z e r r i n g 1134 Thm. 28.81, Vertauschungsr ing [g; 8.2.31)

r = m V(S2 wr S2 wr,.wr S2) ( see e ,g. B; 8.3.2, 8.3.8~,[13; 28 .BA. There fore a l l admiseib le equiva lences a r e equiva lences of S-

systems correeponding t o subr ings o f r. Thus l e t

fo,fl,,,fm be the 2 -o rb i t s o f S2 w r S2 wr,.wr S2 (which

generate the V-r ing F). I t can be ehown (by some computations)

t h a t the products i n r o f the bas is elements ( i n an appropr i -

a te order) ere ee f o l l ows ( m u l t i p l i c a t i o n i s denoted by a):

fo.f, m fO. f uf f,uf = f ( l ~ j $ r n ) , 3 0 3 3 f u f = f a f - 2

3 % 1 3 f o r i ~ i < j S..

'3 1-1 fiafi m 2 ( fo*flt -tfl-l) f o r 1 ~ , i s a . -

We ere going t o prove t h a t every subr ing o f F has a bas ie o f

the form k1 k2 kl

(4 G< fi 1 1-1

h e r e 1 5 k1 4 k2 < .. . < kl m a (1 6 L < m ) . We check t h a t every

bas is element b o f e subr ing R of F i s one o f the elements

of ( a ) . Le t b = . Then the c o e f f i c i e n t s c ( f ) and 'r

G ( f ) o f fi and f ,respa, i n b u b heve t o coinc ide 'r+i r %+l

( l , ~ r < k ) . But

=dfl

c d f i

Thus

i.e., O has the wanted form ( a ) .

On the o ther hand, a l l elements (U) form a subring i n r ( t h i s can be seen e i t h e r d i r e c t l y by checking the d e f i n i t i o n s

o r by consider ing the V-r ings o f subgroups o f S2 w r ,. w r S2).

Thue the admissible equivalences (which have been seen t o cor-

respond t o subringe w i t h bas ie (S) ) are exac t l y o f the form

es prescr ibed i n S3 (1 .e. o f admissible form).

Step 21 Now, we are going t o show t h a t every S-system o f an

S-r ing over Z i s o f the form described i n 93 (Thrn.1. The 2"

proof runsby i nduc t i on on m . The v a l i d i t y of the asser t ion

f o r m m 3 e i t h e r f o l l ows from the r e s u l t s executed by the com-

pu te r o r can be done e a s i l y by hand (checking a l l S-r ings and

t h e i r S-systems, o r aee m). Le t (GO',G1, -0 ,Gm,1 ;B) be the S-system o f some S-r ing R over

z We d i s t i n g u i s h two cases. zm

case ------ 1 r [201e= {2°,21, - m , 2J0-1~is n o n - t r i v i a i (i.e., J~>I).

Case W----- 2 % [2OlB = {2'3 i s a t r i v i a l equivalence c l ass o f B ( jo=1).

I f we de le te i n R e l l q u a n t i t i e s which are contained i n

( i n case 2 (Jo= l ) t h i s are a l l odd numbers, c f .

d e f i n i t i o n o f K/d i n § l ) , and i f we d i v i d e a l l remaining e le-

ments by 2'0, then m s h a l l get an S-r ing R * over ZpmJo

(what e a s i l y f o l l ows from the d e f i n i t i o n e ) w i t h the S-system

' $ 9 ' ) where 01 i s the subgroup Xk o f (G ' ,G' +l.,.Gm,l Jo Jo

~ ( 2 ~ " ~ ) which has the same index k as the subgroup Gi o f

m i 1' ~ ( 2 ), jofi i S m - i (moreover, ( 2 ,2 )GB' M (2i+Jo,2i'+J~)a~).

By induct ion, the S-system o f R ' can be obtained from the

t r e e 4 ( 8 ' ) , i.e., i t f u l f i l s the cond i t i ons ( i ) - ( i v ) o f Theo-

rem A'. I t remaines t o prove t h a t (Go, ,,G ) a re I n eccor- do-1

dance w i t h (11)-($v). We d i v i d e t h i s proof i n t o the f o l l ow ing

p a r t s (A)-(D) t

(A) jo > 1 (case l), then Go= ,=G = ~ ( 2 " ) = Xi* Jowl 1 Because (2O.2 )€B we have Go = Gi f o r l <do. Thus l e t

- The bas ic q u a n t i t y T (R) must have the form

d -1 (1)

b ~.2'.r, T ( l ) = s.0

f o r sore odd n a t u r a l numbers ro= i , rl. ,..r 3 - l Consider -

8 d 0 do- (S-0, ,. ,do-1). Becauee 2 rS + 2 re = ( 2 + 1)2'r,,

T(2Jors) 3

the q u a n t i t y T +l) must be contained i n the q u a n t i t y

C T ( l y Jo t ( 2 rs)

3 i n p a r t i c u l a r (because o f 1rT ) we have 1 + 2

(1) E C f o r

J S = 0, ,m.~o-i, i.e., 1 + 2'o"a a (mod 2 O), i n p a r t i c u l a r

3 S a mod 230, t o r some a&Ti1).

m +1+2'+ k3) f o r some l. Then a t 1 mod 2 2 Suppose B = X32+2( { - f o r a l l srB and we get the c o n t r s d i c t i o n 3. l mod 22 (note

Jo >l) l

Therefore suppose B = X3i+1 ( = G 1 + 2i+2k3 ) . Then

3 s $ 1 + zii2k mod 2'0 whet i s poss ib le o n l y f o r 1-0, i.e.,

B=X1 and we are done. k F i n a l l y . suppose B = X 3 i (= { ( - l ) + 2i+1kf), 13 1 m Then

39 (-i)k+2i+1k mod 2'0 what i s posaib le o n l y f o r e i t h e r j0=2

2 3 o r 1-1, If 1-1, then 1+2 s a mod 2 O f o r some acB a l so i m -

p l i e s do-2. NOW, beeauee of T 5 T(l) H T ( ~ ) , (1)

we have

5 + Zi+'k + 41 mod 2' f o r some (-1)k+2i+1ke~ (2) and

4 l t T (4)

Since m >3, t h i s i s poss ib le o n l y f o r i+l f l ( i f k

i s odd) o r 1 + 2 & 2 ( i f k i s even), thus 1-0 what con t rad i c t s

t o 12 1. (A) i s proved,

J jo=i (1.e.[2Oj"~2~). - case 2) and [21&={21,,.,2

- \ X]'-" -'(97 V t 1 t e n G- = X, = ~ ( 2 ~ l .

element 4 appears 2 m-i-l t imes i n X3 S X3 wh i le the element - - 2 ( ) -2 mod 2' f o r m22 ) appears o n l y 2'-i-2 tirnes. T h i s

con t rad i c t s T (2) T 4 )

. I t remains Go = Xi and we are done.

(B) i s proved.

(C) If d o - l ( [2O4 =f2O)) and 12'4 -{21). then Go depends Gl

accordina --v t o the r u l e s (AV)(= (a) & (b), 53).

Let '0 X3io+p, and ~1nX311+p1 ( P ~ * P ~ P C O I ~ O ~ I ) . If Go'X1 m

are done. Otherwise i t i s s u f f i c i e n t t o show i o s l +l and a (p0=2 4.$ P1 -2) ( then ( i v ) i s f u l f i l l e d ) .

Becauee {2°)and{21) form t r i v i a l equivalence classee o f B, we

have T(l) = Go and = G1*2 =fee2 I a ~ ~ ~ f i n the g iven S-

r ing . Fur ther , 2eT (1 ) HT(l)

g i v e S ( S l T 1 BY de f i -

n i t i o n of the xi's ( c f . 51) t h i s immediately imp l i es io$il+l.

Moreover. if p0=2. I.e. G0=41+2 k 1 O s k < 2 m-i0'2]o then

i0+2 Q ~ 1 ~ 2 ~ ~ o ~ ~ 0 = { 2 + 2 k I o s t < 2 m'i0'2j, what give^

Conversely. i f pl-2. i.e. G1. X34+2, we have t o prove po-2

( i.e. Go'X320+2 ) o r Go=Xi . TO mee th ia , suppose Go=X3io+1.

Then -1 €Go = T(i) &T(2) at T(1) *(-I) = (G1*2) U 0,

But -1 belongs t o t h l s eet o n l y i f io=O (k-0, kg= - I ) , 1.8.

Go=X1, and we s r e dons.

Non suppose Go = X . Then, analogously, 3i0

Consider the elemente o f t h i s aet modulo 2 i0+2 I ( io+2 0 i1+3) 8

i +1 i +2 W get -1+2i0+1=2-(-1)kg-20 k g nod 2 0 , 1.e..

1 +1 i +2 2 0 ( 1 + k 8 ) c 3 - ( - 1 ) k g mod 2 0

Bec~use i O z l W get a con t rad i c t i on f o r both, odd (4

O s 4 mod 2 i0+2) end even k (+ 2 i +2 iO+l(l+k*) 2 r o d 2 0 ) . Th19 f i n i s h e 8 the proof t h a t Go=X310+2 o r Go-XI.

(C) i a proved.

Comparing (A)-(C) w i t h the theorem Ag, one Oase remeins:

a n o n - t r i v i a l equivalence clase. then G ~ E { x ~ ~ x ~ ~ (g. - cond i t i on ( i i ) - o f Theorem Ag) .

By induo t ion end the proof of (C) we know t h s t G1=X1 and

G ~ € { X ~ , X ~ , X ~ , X ~ ) . I t remaina t o exclude G ~ ~ { x ~ , x ~ ) . If

@,€{X j .~4f then, exemining the q u a n t i t y C - T(i) . T(1) *(-1) =

0, HG,*(-1). we See. t h a t 8 h C but 44 C. T h i s con t rad ie ta

(0) i e ~ r o v e d . Thua Par t 1 i a proved, too.

55 PROOF (PART 11).

.- We conslder the q u e n t i t i e s

and

( 1 = T(28) e r f o r e a 2 " r mod 2m end r ~ l mod 2.

We Want t o ahow t h a t the module R generated by these quant i -

t i e s i s an C-ring. Then, obv ious ly by d e f i n i t i o n o f R e the

S-system o f R 1s the g iven System end W are done (except

proving the uniqueness).

However, the S-r ing p rope r t i es § 1 b),c),d) are f u l f i l l e d f o r

R t r l v i a l l y . Thue, e l l whet remains t o show i s cond i t i on a),

nsmely, t h e t T (2) a T ( ~ @ )

i a egain an element o f R,

We proceed by i nduc t i on on w , For m-3 t h i a can be checked

e a s i l y (aee Fig. 3, p, 13 and the Table p, 9 ) . NOW, f o r a r b i t r a r y m, we d i s t i n g u i s h two oasea.

~ a s e .----- I: 1 2 0 k -120, ..., 2'0'~) i s n o n - t r i v i a l ( jo > I). caee ...-.- 2: [2°3,={20j, ( j0=1).

Again, es i n Step 2 of p a r t I of the proof (§4) (de le t ing jo-1 U z2,,/2' end d i v l d l n g the remaining e lenents by 2 3 0 ) ~ we 8-0

aan aeeume (by induct ion) , t h a t the q u e n t i t i e s

T f u l f i l t he needed C-r ing p r o p e r t i e s , I t ( zm-l)

remains t o show t h e C-r ing p r o p e r t i e e i n which T(1) 1 s in -

volved, i.e, we heive t o show tha t , for t U 1 mod 2,

f o r a l l X Z T(t) - - zm can be repreeented as a sum - (1.0. a~ an element o f R ) .

To s i m p l i f y t he no ta t i ons , l e t

A t f i r s t we obeerve, t h a t , w i t h respec t t o m u l t i p l i c e t i o n r,

t h e q u a n t i t i e s K e i

behave l i k e the fmOl ( I - O , l , ...,m)

d e f i n e d i n §4, p.17. Thersfore, i n the f o l l o w l n g , we eome-

t i a e s uss t h e m u l t i p l l c a t i o n t a b l e f o r t he fl, i d e n t i f y i n g

= 12m-1Z fo w i t h K = M , fl w i t h K . -*, fk w l t h K m-k 2'" - 2 2 - I~z"'~ 1 X~1,3.,.2~-13 . ,, f, w i t h K -0 = f ; \ t ~ ,m I ~ o d d ) .

Note, t h a t fo+...+f - K J +,+K 2m- l 1 s an element o f t h e '"-Jo 2 0

module R generated b y ( O ) . We beg ln w i t h

...r G =X1, Thus K Z 0 t ... j K end we have 3 7

2J0-l

h S T ( 2 ' r L = l T ( 2 8 r ) I *T U f o r e ?Jo - (r odd).

Moreover, f o r k - a - J o + l we have (beoauee K =Jo- l = f k ) '

- - m rn-1

- 1 . , * . C 7 C * *C \

C~~~~ ( j o= i , i.e.[2Te ={2°f) : We d i v i d e the proof i n t o seve-

r a l ateps (A ) - (D ) dependlng on the group Go. Note t h a t

(A) Go - Xi : Then

* T l 2 s r l

C R f o r 821 (r odd), end - = I T (Zer) ' *LU

= { 1 + 2 3 + ~ ~ 1 o L A C 2"j'23 ( 0 f J L 0-2 f ixed) . Th ie impl ies, by the oond i t ions o f the 'Theorem A v ( i v ) , the

f o l l ow ing s t ruc tu re o f the groups GI, ....G 8 J

'1 '3cq1+2 w l t h j-lLqlCm-2,

'2 ' X3q2+2 w i t h ql-1 b q 2 501-2,

coneequsntly J-#&% f o r o(=1,2, -,J.

1 Moreover, 2'. 2 . ,.., 21 and a lso 2 ( I) nuat bs aingle-ele-

ment equivalence c lasses of 8 , i n p a r t i c u l a r we have K ,j+lCR

and K 23+2 ?-*t K2R-1

€ R -

Now, l e t t g 1 rnod 2 i n d z.2.r w i t h odd r end er{o.l.,.,

m-1). ~e want t o ahon €R end d i s t i n g u i s h severa l T ( t ~ " T ( z ~

osees (B1)-(B4) :

( ~ 1 ) ~ + Z Q K (i.e., t + z ~ i mod 2). T h e n z h e v e n a n d MLiLW 2O

t+ueK 2O

f o r e i l 1 ur7 ( 1 . Thus

(82) t + z e K w i t h 1 L k 6 j . Then z muat be odd end wo heve .+yur 2

Because Gk = X 3qk+2 w i t h qk+k - L J and there fo re

k qk+k+2 m-qk-k-2 T 4 2 +2 . (Zk)

-1 3 , the quen t i t y A - i s a sum o f q u a n t i t i e s o f the form T

(2kZ "

w i t h odd r (i.e. a eum o f o r b i t e of Gk on K k ) and there fo re n L

en element o f R.

(83) t + z a ~ UCLU 2 j + l *

Then A ( a s i n (82)) equale K 2j+1

(84) t + z eK w i t h k 2 J+2. Then A (as i n (82)) equals 2

-

T h i s i rnpl ies (by the cond i t iono Thm.A1(iv)) the f o l l ow ing

e t ruc tu re of the groups G1....,6 t 3

end w i t h j -k <qk f o r ke{ko,,.j) provided t h a t

1 3 Moreover, 2'. 2 „., 2 and (beoauae o f qk- i~ l )a lso 2 3 + 1

must be eingle-element equivalence claeises of 9, i n p a r t i c u l a r

we have K *J + l 6R and K j+2 f W t Kpm-i i R. We proceed aa

i n case (B).

Non, we oonclude e a e i l y r If t+r i K w i t h l G k g j , then 2

A + B ( o r and B', reap.) i a s aum o f o r b i t s o f Gk on K m I -

2

(i.e. a sum of quant i t iee i o f the form T w i t h odd r), (2kr)

provided t ha t Gk = Xjq +% ( o r Gk = X , resp.). Therefore k

A + B = A D + B m ER. 3qk

m m - -

I f t + z dK *j+18

then - A = B = K - *j+1' R . - 1f t + z a K w i t h j +2 t k f m-1, then I. Am B = K +2?-.tKZm-1 ER.

2 23 - I n every Gase8 A+B-bR.

Anelogoualy we can ehow C + D e R (use t-z, C, D, reap., i n - - - stead of t+z, A, B, reep.). Thua - A + g + g + g ~ R and we a re

done.

A s i n (C) we get now

i n p e r t i c u l e r , j+2gqk+k+1. We prooeed sa i n (C) and get f o r

z € 2 \{o] snd odd t: 2 m

(01) t+2 K MMu

e) If z r K w l t h k 2 j+1. then 2 2

b) If SK w i t h 1 g k g j end G k = x 3qk+l' then 2

o) If z€K (1 i k $ J ) and Gk = X . then 2 3qk

(02) t + Z € K k, 1 t k - m - 1 . Thsn - 2 {(-1)2t+2j+12 ] %=0.1. 2mod -Io1

T(t) 'T(z) J

Few computetions show t h e t

where A,B,A' ,Bn are as i n cese (C2) end

C - I( t - ~ + 2 ~ + ~ ) + 2 3 + ~ A I A~A?, D D.{-( t-z+2 J+i) +2J'2 ;1/ xc AI 1

Ce.{(-1) ( t -z+2 j+1)+23+2~ 1 l d A )

0.. (-1)a (-t+2-2J+1)+2J+2~ I A C A).

Now we can proceed e x e c t l y ae i n oase ( C 2 ) snd ob ta in

Th is f i n i ehee the proof o f Case 2 end, moreover, tne proor

o f the theorem,

Remerk. For a g iven S-eystem, we d i d no t show the uniqueness - of the corresponding S-ring, T h i ~ , however, f o l l ows from the

d e f i n i t l o n s (end known S-r ing p roper t ies ) , because d i f f e r e n t

S-ringe cannot have the eame basic q u a n t i t i e s T ( 1 ) , ~ ( ~ 1 ) ,

T , whioh, i n turn, are un ique ly character lzed by Go,G1, (2m-1)

and by 8 (due t o the Pa r t I o f the proof) .

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NoL, Najmark u l , Karpenko 30-10

p r , 6 0 - l e t i j a O k t j a b r j a 9 SU-327038 N i k o l a j e v , ob1.38 SU-117312 Moskva

R. Pöschel I n s t i t u t f O r Mathematik AdWdDDR