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

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

P-MATH- 14/81

Michail Ch. Klin. Nina L. Najmark. and Reinhard Poschel

Schur-rings over Z2,,,

Berlin

Keywords

S-ring (Schur -r ing )

Permutation group

AIüS SubSect olassification (1980)

2OCO5, 20-04, 20B99

Received April 15th. 1981

Summary, The f u l l deecript ion of the s t ruc tu re of a l l so-called S-ringe

(Schur-rings) over Z provides much information on permutation groups n

containing Zn a s a regular subgroup (I, Sohuris method). I n the pre-

sent paper some experimental material f o r the inves t iga t ion of S-rings

over Z i s presented,namely the l ist of a l l S-rings over Z f o r m 4 5 2m I 2m

worked out by computer. The program f o r these computations 1s sketched.

Moreover, the f a l l deecript ion of a l l S-rings over Z ( m a r b i t r a r y ) 2m

1s announced. Together with previous r e s u l t s of the authors, the struc-

t u r e of S-rings over Z ( p a prime number) i s now known. pm

Zusammenfassun~. Die vol ls tändige Beschreibung von sogenannten S-Rin-

gen (Schur-Ringen) iiber Zn l i e f e r t wesentliche Informationen über Per-

mutationsgruppen, d i e Zn a l s reguläre Untergruppe enthal ten (Methode

von I, Schur). I n der vorliegenden Arbeit w i r d experimentelles Material

zur Untersuchung von S-Ringen aber Z vorgelegt und zwar die m i t Hi l fe 2m

e ines Computers berechnete L i s t e a l l e r S-Ringe fiir m 4 5, Das dabei

verwendete Rechnerprogramm w i r d kurz sk izz ier t . Außerdem w i r d d i e vol l-

ständige Beschreibung a l l e r S-Ringe über Z (fiir bel iebiges m) ange- 2m

geben. Beriiaksichtigt man frühere Resultate der Autoren, so i s t damit

d ie Struktur a l l e r S-Rings über Z , ( p ~ r i m z a h l ) bekannt. P

There are two old ways of investigating permutation groups containing a regular subgroup of the eame degree. Beside the method of group characters one also has the less-known method of I. Schur /Scu which is much more elementary and produce8 nevertheless similarly deep results, Schuras idea has been taken up by R, Kochendtirffer and H, Vielandt, and can now be presented ae follows, Given a permutation group containing a regular subgroup H of the dame degree, we assign to it a

subring of the group ring Z (H) , a eo-called S-ring (Schur- ring ovgr H), For basic results in this respect we refer to f~ch],[~ie u,[~ie 2],[~o ~],(Ko 2],fld(cf, also DtJ/Ka (Chapter 8 ) l ) .

Later it turned out that S-rings are in close co~ection not only with permutation groups but also with b i n w rela- tions, which enablee to solve some graph theoretical problems by using S-rings (see e.g. h6/~q,?,[~l/~t! I.~,[IU/P~ g).

In this paper we are concerned with S-rings over the cyclic group Zn (in particular for n=2m)e The f'ull descrip- tion of (the structure of) all S-rings over Zn provides much information on permutation groups containing 2, as a regular subgroup and yields answers e.g. to the KBnig problem and to the ieomorphism problem for the eo-called circulant graphs (cf. [PG/K~ (Chapter 8 ) J ) , Because of theee and other possible applications the following problem is of interest:

Problem: Determine and describe explicitly all S-rings over Zn .

This problem was solved for nmpm (pC2 a prime number) in [Pu and for n=pq (pCq prime numbers) in [gl/~G 1. (cf. also [~i/Ka (8,4,fe)]). In the powers of 2 were excluded for n because in this caee the prime resadue claee groups

~ ( 2 ~ ) modulo 2m have a more complicated et~cture (they are not cyclic for m 3 ) .

Recently some effort6 have been made by Ja,Ju, Go19fand (AN SSSR, ~oecow) and by the present author8 to derive the structure of all S-rings over Zn (n arbitrary) from the etructure of the S-rings belonging to the corresponding prime factore of n, Should these investigations lead to eatiafac- tory results, the above problem would be traced back to the remaining case n=P. T h i s case is attacked in the present

paper.

How to describe the S-ringe over Z ? In order to get 2='

a better feeling to treat thie problem, we ueed a computer for determining all S-rings over ZP with mf5, By this, the description of S-ringe has been based on that of the eo- called S-system8 introduced in h5], N,L, Hahark worked out the computer program and presented the list of theee S-rings, Examining the list obtained in this way we hoped to find a general conjecture on the structure of S-ri*s over =P* We did find a reeonable conjecture, which hae recently been proved by Ja,Ju, Gol'fand, M,H, IUin and B,L, lfaherk,

Following a suggestion of M,H, Klin, V,A. Zai8enko (METI, ~oecow) worked out mother computer program for the determi- nation of S-rings. His algorithm uses lese S-ring theory but givee much more information on the computed S-rings (e,g. it

a180 determines their automorphiam p- P group8). p However, concer- ning S-rings over 2, he managed to treat only earller n'e

in a reasonable computer time (e.g. Z2rn up to m 4 and by a recently modified algorithm also for m+),

p

In 91 of the preeent paper we introduce some notions we . need from S-ring theory, In $2 we present a conjecture on the etructure of S-rings over ZZrn (theorem 2,3), In 93 we give a short survey on the algorithmueed for the computer program. In 54 the list of all S-rings over Z2= (l g m ~ 5 ) ie given,

The aim of this paper is tbreefoldt to present some ex-

perimental material for the investigation of S-rings over Z (m4 S ) , to announce the full deacription of all S-rings zm over Zp (m arbitrary), and to demonstrate the "interaction between man and computerw in solving pure mathematical pro-

blem~, We have the following procedure in minds Ueing re- latively few information8 on the S-rings (and therefore a

more or lese complicated program) the computer provided auf- ficiently many examples, whioh in turn gave a hint to des- cribe the structure of all S-ring6 under consideration, Moreover, our algorithm used for the computation ha8 an im- portant adv&age: it admits 'theoretical progressH, i.e. the more we know the structure of S-rings under considera- tion the more the algorithm can be modified into a better version, (The algorithm checks posaible candidate8 for so- called S-aystema which in turn characterise the S-rings. Therefore every further (theoretical) information on the S-ring structure reduces the number of these candidates and shortens the algorithm,) By thia, e,g. ~omconjectures on the structure of S-rings over z~ have been checked also for m=6 and m7, Of course, we get the best version of our algorithm if it will be modified by ueing the structure theorem obtained (in 52),

ACKNOWLGDGUENTS, Our thanks are due to Ja.Ju, Go19fand and V.A, Zd8enko for stimulating disousaiona and remarks, and to V.A. Zdzenko also for the permission of using hie computer results,

$ 1 s-=as AND S-SYSTEMS

No t ione and Ho ta t ions ,

Let n € H (where B denotes the se t of a l l na tura l

numbere) and Z;{O,~, . . . ,n-l). Producte and sume of elements

of 2, a re taken modulo n ( i , e m , we work i n the residue

claaa r i n g z,), i n pa r t i cu la r , we define f o r T , T 9 = Z n ,

X&,: T + X := i t + = l t aT f T + T9:= {t+tp 1 t e T , t 9 ~ T 9 j -

The grime --------.W-- residue c l a s s .---- g g o , ~ ~ of a l l numbers qszn(nc l) re la -

t i v e l y prime t o n i e denoted by

~ ( n ) = [ q c ~ , 1 g.c.d. (q,n)=l).

1,2 L e t n e B a n d l e t

be a d i s j o i n t union of subseta (i.e. a p a r t i t i o n ) of Zn with the following propertiee:

( i ) ~ ~ = { 0 3 ,

( i i i ) For yeTk, the card ina l i ty p i 9 j of

T - T ) depends only on k but not 3 on the choice of y ( ~ & i , j , k f r ) ,

Remark: (iii) ia equivalent to

(iii) * If xeTi, ycTj and x+ytTk , then all ele- ments of Tk appear in Ti+TJ with the

k same multiplicity pi S 3 (i.e., Ti+Tj is the

union of suitable S with corresponding

multiplicities pk i d ).

For z eZn let T(z) be the uniquely determined Ti such

that zrTi. The system

with the above properties (i)-(iii) is called the system of

all basic ---W- quantities -W----w-- of an S-ring over 2,.

S-ring8 were defined e.g. in D i e 11 or [Pa/~a]r An S-ring

with basic quantities T(z) (zt~,) is the eubmodule of the

2-module 2(Zn) generated by the T(,)*s (sometimes there is

used another ring R instead of 2) . Because of the proper-

ties (i)-(iii) this submodule is a subring of the group ring

<z(z,) ; +, 0 ) of Zn over 8 . Every S-ring is uniquely dew

termined by its basic quantities; thus, for ehort, we will

.W--- over Zn . speak of the Soring s~(T(~ )>zeZ, ----

l .3 Examples. Every permutation group G f yn (yn= the full aymmetrio group over 2,) containing

(where a": X *x+a (xe~,))

determines an S-ring over 2, in the following w a y : Let

Go = { g ' ~ 1 og=Of

be the s t a b i l i z e r of the point 0 and l e t

Then (by a r e s u l t of I. Schur

i s an S-ring over 2,. he two S-rings s ~ = ( T (. ))scZ_ w i t h LA

s and s l"{~(z))zLz~ w i t h T o f o r %+o

are c a l l e d t r i v i a l . We have ------I

S-rings can be character ised by so-called S-syeteme

(introduced i n (Pal, cf. a l so [ P ~ / K ~ , / I ( ~ / P u v). Let

be an S-ring over Zn and l e t

be the s e t of a l l d iv i so r s (h) of n and do=le I f , i n par- 1 t i c u l a r , ~ i . 2 ~ we take ~ { 2 ~ , 2 ,..., 2m-13, di=2 i . Bow, r e

ass ign t o each S i t s -g- S-sgstem g-m-

defined as follows:

0 6 i 6 m-1 , ( s t a b i l i z e r of Tcdi) i n P(n) ) ,

where 5(,):= q is the so-called -W--- t r a c e q€P(n)

Obviously, ~ ~ ( 9 ) is a subgroup of ~ ( n ) and 8(S) is an

equivalence relation on D. Moreover, ~ ~ ( 5 ) has to contain

stab(di)={atP(n) 1 dia 3 di (mod n)j.

A sequence C = (Ao,Al,... ,Amg1; 8) is called an S-sxstem .-m ---- over g-0g 2, if there is an S-ring 8 over 2, suoh that E(S)-C.

Every S-ring S is fully determined by its S-system

because one uan ehow without difficulties that we have

([Po/K~ (8.4.9)],fiie 1 (23.9(a) )l):

T(di)= ( d i m 9

*(z) m (di)q for i=diq with qc~(n), z~z,.

Of course, not every eystem C-(Ao,.. .,Am-l;8) is an

S-system. Therefore the problem now consiats in finding a

characterisation of all S-systems over 2,. As mentiohed in

the introduction this problem was solved for mprn (pf2)

and n=pq (pfq prime numbers). In the next paragraph we

shall treat this problem for 11rp.

$2 THE STRUCTURE OF S-RINGS Om 2m

The computer reeulter given i n the next paragraphs together

with someS-ping theory had been the background f o r the con-

jecture (proved i n the meanwhile) which we a re going t o for-

mulate - ae a theorem - i n t h i e paragraph.

F i r e t of a l l , we need a description of the subgroups of

~ ( 2 ~ ) - the prime residue c l a s s group(cf. 1.1)- because such

subgroupe form the S-syetems.

The subarou~s of ~ ( 2 ~ ) . 10 a 2,

She following s e t s G1,G2, . . . S Z p are precisely G

sub@.pups of ~ ( 2 ~ ) , m 2 2 ( a l l nwbers a re considered mod 2 9 r

i .e. , -1 and 5 generate the whole (mult ipl icat ive) group

~ ( 2 ~ ) ~ c f . e . g . h d . Therefore the subgroups of ~ ( 2 ~ ) a r e

generated by the following elements: Q ~ ~ + ~ ~ ( - ~ , 52i)

A 9

b) Of course, the G 'a depend on m; therefore we could 4 write inetead of Gj for the subgroups defined in

G~ 2.1. But thia is (in some sense) euperfluoue because we

have for mSrns:

(m') modulo 2m G!') . Gj the characterization of G by generating element8 (see

j remark a)) doee not depend on m .

Now we shall describe S-systems of S-rings. Af'ter descri-

bing epecial equivalence relations in 2.2 we are ready to

formulate the characterization thsoren 2.3.

2 Let ~,={1,2,2 , . . . ,2 m-lJ be the set of a11 divisors ( #2') of 2' (re 8 ) . An equivalence relation 0 on D, is

sible if every equivalenoe class is of the form

for some 0 - 5 i , l e m , if (d,ds)cO (ds ds) then the

whole interval1 [d, d9] = {dnc D, I d 5 dn 5 d9{ belong* to the

same equivalenoe clue of 8. There are exactly F1 admlasible equivalence relationa.

Characterization theoren for S-erystem over Z2n

For subgroupe 4 (O~ihm-l) a ~(2') (m221 g& =g eauiva- - lence relation B Dm , - the s~atem C=(A~, ...,~-,; B)

is an S-system (cf. 1.4) if and only if the followinq con- -- -- -- - ditione ape satisfied:

( i ) O is an admissible equivalence D, (cf. 2.2). il il+l i + i

( i i ) { 2 ,2 , . . . , 2 3 2 pp ,equiv~1ence c lass

of 8 with more than one element, then - Ail= .= Ai1+i2' O 1 Ail-l- - and

( i f i l = O o r il=l then delete the conditione where

negative indices appear) ,

( i ~ ) l$ jLm-2. g Aj= G w + l - Or rj' G3i+3 - fo r some

O s i f -2, then

A ~ - ~ ' { ~ ~ $ ~ I B ~ ~ + ~ I 0 1 8 6 i + l { u . ( ~ ~ ~ ~ ) 0 5 r 6 i f ;

for some 0 i S m-2, then If '3= G3i+2 - - +01 t{~ l fu{~3s+21 0 5 s ~ i + l $

Remarks: a) Theorem 2.3 was formulated as a conjecture but

i t was proved recently by Ja.Ju. Go19fand and the preaent

authors. The proof w i l l be published elsewhere.

b) If $f 4 and A ~ c { Q ~ ~ ~ G3i+lr G3i+2 J then i Jrn-2-j

j 3+1 and 2 ,2 ,...,2 J+i+l are pairwise inequivalent with re-

spect to 8.

c ) For every given admissible equivalence re la t ion 8, every

S-systm (Ao,. . . ,Am-l ; 8) can be described by a directed

path i n a t ree (rooted digraph without aycles), We give three

examples:

To a directed path starting from the top (the grootn) and

passing through vertices with numbers 1=i0,i1,i2,i3 we

G G G 8 8) which is an S-system assign *he system (Gi3' i2, ill 1

indeed. The corresponding S-ring will be denoted by Si . 3 2 1

Here a directed path with vertices

givea the S-systems

The corresponding S-rings will be denoOed by I

d) T b graphs ( t r e e s ) considered above can e a s i l y be con-

s t r u c t e d from theorem 2.3 ( t o each e q ~ i v a l e n c e c l a s s of 8

corresponds a l a y e r o f the d i r ec t ed graph), We w i l l not go

i n t o d e t a i l s here because these r e e u l t s w i l l be published

elsewhere ( the proof of 2.3 included),

e ) A s mentioned i n the in t roduc t ion the s t r u c t u r e of a l l

S-rings is now known f o r a l l prime powers nSpm (by theorem

2.3 and [PU), Theorem 2.3 oan a l s o be used f o r a satisfac-

to ry so lu t ion of the isomorphism problem f o r c i r c u l a n t graphs

over Z y ie ld ing a r e e u l t s i m i l a r t o bd&n9s conjecture P ( c f , [U/P~ 2]), There i a one more problem worthy t o be in-

ves t iga ted , namely t h e descr ip t ion of a l l automorphism

groups of S-rings over =2m

(giving a l l so-called 2-closed

[wie 27 permutation groups containing 2" ). With theorem 2m

2,3 t he re a r e no se r ious d i f f i c u l t i e s t o solve this problem,

93 THE COMPUTER ALGORITHM

In -this paragraph we only sketoh the algorithm used for

the construction of all S-rings over Z (because 2.3 gives 2m

a much better algorithm).

The program for this algorithm was written in FORTRAN

(by #.L. arma ark). The computations were gerformed on a

computer of type E0 1020. The determination of all S-rings

over Z2rn for 3 .f m f 5 required nearly 2 houre computer

time (the program of V.A. ~dxenko mentioned in the intro-

duction required more than 3 hours for the determination of

S-rings and their automorphism groups over Z2, for -4 ) .

The algorithm used here ie baeed on the obeervation that

every S-system determines an S-ring (1.5), Therefore all

tlcandidatesn for S-systems are determined and then the S-ring

gropertiee (1.2) for the corresponding S-ring will be checked.

Now we shall sketoh eome eteps of the al~orithm:

Working in Z 2m

all arithmetic is done modulo 2*.

(1) Construction of all (multiplicative) aubgroups

of P(P), cf. 2.1.

(2) Construction of all equivalence relations 8 of the 1 set D ={2°,2 ,..., 2rn-1) of all divi~ors(#2~) of p.

We consider 8 as the set of its equivalenoe olaeses

(i.e. as a partition of D).

The following stepa will be done for every 8 (main cycle I).

(3) For every equivalence claas K48 , the greatest ele- ment (w.r.t. the natural order of numbera) d=d(~)

of K will be determined. For this d, all eubgroupe

of ~ ( 2 ~ ) containing the stabilizer

B(K):~(P(~~))~s={x~P(~~) l dXmd f will be determined. For short we ehall call these

groupe admiesible W--------- aubgroupe (war. t. K&).

(4) To every K40 an admissible subgroup F(K) will be

assigned.

Remark: The mapping K )--+P(K) provides poaeible candidates

for the S-system of an S-ring S with 8(~)=8 and Ai(S)=

=F(K) for 2 % ~ 4 note that B(K)~A~(s) (in case 2 % ~ ) is

obviously a necessary condition for the Ai (S), of. 1.4.

The following etepe will be done for every assignment

K eF(K), KIO (main cycle 11, where F(K) runs through all admiseible eubgroupe). Moreover the following etep will be done for all m e (main cycle 111). .

-- -- -- p ---P----

- - - - - - - -

t ~2 0. For every element 2 % ~ the orbit*

of the group F(K) on 2r~(2m) will be determined.

"i To every 2 EK (OLiCt), an orbit Qi=Q(zi) ri will

be assigned such that s0=2r0. Thua, to every K re

can assign the set

(63) 17

Remark: Q(K) is er candidate for a basic quantity T of an

S-ring S with P(K)=s~~~(T) (cf. 1.4).

The follalwing etepe will be done for every aesignment

K M Q(K) aa given in (5), KrQ (main cycle IV, where Q (K) runs throlugh all unions of poseible orbits Qi).

p -p p

(6) F r every Q@), Keg, the set of all m-oalled adjoint

q antities T-Q(K)q (where qe~(2m)) rill be computed.

L t S be the set of all these quantities: I Remark: $ is a candidate for the set of all basic quantities

ing, In the following steps the property of being an

11 be checked4 more preciaely, we check the property

1.2 (iii) for all Ti,TjcS (= main cycle V). 1 revery two elements T and 91' of S the set

T + T1- {t+tt 1 ttT, t' t ~ ' f

11 be computed noting also the multiplicities of the

ements (i.e. how often appears every t+t' ).

(8) Id;

i~

t1.e

If

will be testedA(property 1.2 (iii) ') whether T+Tt

the union of some suitable quantities of S such

tk.at all elemente of a TneS appear in T+T1 with

same multiplicity (which may vary with T ~ ) .

the teat ie positive for all (not necessarily die-

tinct) T , T t e S , the qufn tities of S will be printed

or,t.

Remark: By ( 8 ) , f o r every p a r t i t i o n 8 the s e t of a l l S-ringe

with 9(S)=9 w i l l be pr inted out.

As mentioned i n the above remarks, the program contain8

f i v e main oyoles varying the 9, (1)

the assignment K F(K) , Keg, (11)

the Keg, (111)

the assignment K Q(K) , (m the elements T , T i € S . (v)

04 IJST OP S-RINGS OVER Z p ( 1 m 5)

I n the following a l l S-rings over Z2m ( I j m 5) w i l l be

l i s t e d . h r the basic quant i t ies T of S-rings we use e i t h e r

the notation

T = {a,b, ..., c i o r T = a,b,...,c (underlined).

The notat ion Si I f o r the S-rings is motivated by 1 2 " '

the deecription of corresponding S-systems (see Remark a ) af-

t e r 2.3).

S-ring6 over Z2 (n-2, m l ) :

There I s only one S-ring over Z2:

S, -<Q, L>.

4 2 (S-ringe over Z* (n=Z2, m=2)r/ _ _ . c

There are three S-rings over Z 4 (two of them are t r i v i a l ) .

e(s)=N20$,{2~J.I: s1 =(Q, U, g >

s2 - (0, 1, 2, 2) ~ ( ~ ) 4 2 ~ , 2 l f ] 8 Slrm (Q, 1*3*2.).

S-ringe over z8 ( n d 3 , -3): 1 +

I n the following l i e t we b i t the basic quanti ty T(~)={o)

because it is contained i n every S-ring.

Remark: S52 curd S1. are the t r i v i a l S-rings .

'+rings over Z16 (n=24, m=4) r W

There a r e 36 S-rings over Z16 (again we omit T(0)={~]).

T(8) T(22)q T

(211q r

T ( 2 0 ) ~

Num- ber

lota- t ion

---.-----------------------

Sll$

'112

'121

'122 1.3.5.7.9.11.13~15

'131

'141- L

'152

u . 1 2 , j.7.11.15 l

1 2 :S231 l

1.3a9.11, 5.7.13*12 - l8 ? S341

l9 is41 + 20 S412

2,6,10,14

2.10, 6.14

-9 2 6 10.14 - 2.14, 6.10

- 2 , $ , 2 , 1 4

2.6.10.14

2.10, 6.14

-9 2 6 10.14

2.14, 6.10

2 , 5 , 1 0 , 1 4 - 2.6.10.14

2 6 10.14 - 9 - 2,14, 6.10

2.6.10.14

-9 2 6 10.14

' 2 1 4 6 1 0 L, L

I

2 1 I

2 2 Sqql t I

1.7m9.15, 3,5,11,12

4.12, 4, 12 0 - 4.12

4, 12 0 -

4.12

3 , g 4,12

4, 12 4.12

9, 12

4.12

8 g 8 - g

8 L - 8 - - 8

8 - 8 - g - 8

8 -

4.12 4.12

4, 12 0 - 4.12

4.12

9. 12

4 12 A

- 8

8

8 - g 8 I

7

- 8 -

g 8 0 . 8 0 - 8 -

I

Remarkt The S-rings number 28 and 37 are t r i v i a l .

N u - ber

23

24

25

26

27

28

I @l)9

2.10, 6.14

2 , 4 , 2 , 1 4

-9 2 14 A 6 10

g, 6, 2,

N o t e t i o n

'521.

'522 ,

S552

'641

S741

s852 2 3 ~cs)={+o{ ,{2lJ ,{2 ,2 32:

T(22)q

4.12

4, 12 &A2

4.12

3, 2

T ( 2 0 ) ~

U, 3.11,~ 5.131 7.15

X, U , 9.15, 11.13 '

-U,, 3 d , , 3d, U

- l, 2, 3, 1, 2, &L,&21&2

'(8)

8 - - 8

E

8 0 - - 8

E

,29 30

S1ll'

S21, 4,12,8

----7---------------------

1.3.5.7~9,11.13~15 - 1.5.9.13, 3.7.11.15

1 2 @ ( ~ ) = { { 2 ~ f ,.I2 ,2 j,.123f]:

2,6,10,14

----T 31 ISl1' 2,6,10,14,4,12

---------------.---D

1,3.5.7.9.11.13.15 - 8

1 2 3 e(s)={{20] ,{2 ,2 ,2 jf: ---- ------I----------- -

32 /Sl1., 1.3.5.7.9.11.13.1~

- - --

2,6,10,14,4,12,8

@(S)= {12°,211, k2) ,42313: ----v--------------------

1.3,5,7~9~11~13~15~2.6110~14

1,3,5,7,9,11,13,15,2s6~1Qr14

t

4.12

- 4, - 1 2

33

34

- 8

g S1ml1

S1m2& 2 3 ~(s)={{20,2Y ,12 12 Jj :

4.12.8 ..LoII~-.....llLo-..o---.o- - ------- -

35

0 1 2 Q(s)={{~ ,2 ,2 f , k 3 f ) r $l# X# 1,3,5,7,9,11,13,15,2.6,10.14

----------p----------. - p--- - -

1.3,5,7.9~11.13,15,2.6,10~14.4112 W 8 36 Sl"= 1 2 3 8(5)={{2O,2 ,2 ,2 !l:

P- -

1.3,5,7,9.11,13,15~2.6,10.14~4~12,8

o-o.---o-------t

1

37 S1m

S-rings over 232 (-25, -5)s

There are 151 S-rings over Zg2.

5k- bel:

23

24

?dj 16 16

4,12,20,28

Nota- tion

'1521,

*(2qq $(231q

g, 24

4.20, 12.28

-

--- p----- --

' ( 2 0 ) ~

1,395979

-- 8.24

---- 16 -

- --- -- --

T(21)q

A* 2 18 L) 6 22 10.26, 14.30 '

~ ( 2 4 4

16 -

- 16

J&

Elum- ber

118

lig

120

1

~ ( 2 2 ) ~

4 28 12.20 -J--'

4,12,20,28 ----

----------.-.-----..--.

6.12.20.28

T(23)q

8.24

8 , 24 -

L -P-

8.24.16

- - -- -

~ ( 2 9 9

2.30, 6.26, 10.22, 14.18

& 6, 2, 18,18, 22, 26,x -

Nota- t i o n

s974i

% Q ~ u

%852

P-

-- - -- T(20)9

.U,, 3 a , . 5.11, U , U., 19.29>, 21.21, 23.25

,U,, Ja, 5 a , , 7.25, 9.21, 11.21,

ia.12, U,T,1,2* , , - u,21,22,22, 21,29,3i

P-

Remark: The S-rings with numbere 120 cuid 151 are trivial.

%rings over z 6 (and partiaiiy over z 7) &so wsre 2 2

investigated by the algorithm described in 03. We mention here

#

only thot there ars 657 5-rings over Zs4 (m=6), 538 ~f

'(231~

whioh have the trivial equivalenoe B(s).

Tz4) T(20) t

.--- 144

-W„--

145

~ u m 4 Nota- ber tion

1j 1 2 ~ 3 123 243fl ~ii~=ji~!A~-.a ---- &---I -- sl~llb

o ,l--.l---l„l----„--.--.-----------------------. pij 122 23 24ih

[113r5r0 ..,3ip2,6plo,o oop30f 4.12.20.28 8.24.16

146 S1qn {lp3,5r ...P 31p2,6plo,o . .JOJ 4.12,20,28,8,24~16

elslp A20 2lf b2 23f k4fj , „,-1---1--.1---1-.--.--.----------------.------4.--

__._,._

147

148

- - - , V „

149

„-9.-0

150

,--„„

151 C

M ~~~,{1,3,5,... 31,,6,10,...,30 4.12.20.28.8.24

eis_r-iia!,z!,~?l,ia?i~1~4!f~ ------ -...---------T--

slnll L 8 24 {1,3,5 #... p31,2p6p1~, ...,30~4,12~20,28J

sl"21 s, 24. 16 i6

elsl- 1-20 2l 22t 123 24ff~ - -- --1--1---1 --1-- ---,--,-----------------.------- S1mla {lP3,5, ...r31p2,6p1~p ..., 30~4~12,20,28J 8.24.16 elsl= Ji20 2l 22 23f h4f - -, S1w1

„1-.11„1„.1---------------.-----------------.---

{lp3,5ro ..,31,2,6p10, ... p30p4p12p20~28~8p24~ eisl=i12~ 2l 22 23 24!]: - -,-„1„1--1--1--„---,---.------.-----..--...------..-

Sinn {1,3,5p...r31p2,6p1~ ,... p30p4,12,20,28,8,24p16f '

[~a] H.IUSSE, Zahlentheorie. Akademie-Verlag, Berl in 1963.

m / P ö M,H,KLIbl and R.PÖSCHEL, The K6nig problern, the iso- morphierm problem f o r oycl ic grapha and the charaoteri- s a t i o n of Sohur r ings , Preprint AdWäDDB, Z I U , Berlin, März 1978, (11 rahortened veraion appears i n the Procee- dinga of the Intern, Coll, on Aigebraic methods i n graph theory(~oeged 1978). Co11 , 1Ylst. 300. Jhoe Bolyai 21.)

Dl /Pö 21 BII,H,KLIH and R,PÖSCHEL, The ieomorphism problem f o r c i r c u l ~ n t digraphs w i t h pn vertioee. Preprint P-34/80 ZIMM, Berl in 1980,

[KO 11 R.KOCKEM)ÖDFER, Unt ereuchungen Uber eine Vermutung von W. Bupneide, Sohrif ten des m a t h , Serainara und des Instituts f ü r angewandte IYIath, der Universi tät Berlin, Band 3, Heft 7, (19371, 155-180,

D o 27 R . I C O C ~ Ö R F F E B , Lehrbuch der Gruppentheorie unter be- eonderer Berücksichtigung der endlichen Gruppen,

, Leipeig 1966,

[PÖ] R,PÖSCEEL, Untersuchungen von S-Ringen, inebesondere i m Gmppenring von p-Gruppen. U t h . Bschro &(1974) ,1-27.

[ P U / K ~ R.PUSCHEL und L.A.uuZNIN, Funktionen- und Relationen- algebren, DW, Berl in 1979.

(Sch] I,SCHUR, Zur Theorie der einfach t rans i t iven Permutw tion%nippen, S ,-B. Preuße Akad, Wioe., Jahrgang 1933, 598-623.

Ba] O.TAMASCHICE, Sohur-Ringe. B .I,-Hoch~chUltaschenbficherei

(735a), Biblio-, I n s t i t u t , Umnheim-Wien-Zürieh 1969.

[wie 11 H.WIlUUD!P, Irinite permutation g r o u p ~ . Academic press, Bew York and London, 1964,

[wie 2] H.W,W~ELAM)T, Permutation groupe through invar iant re ia- t ione and invar iant Functions, Lect. Ohio S t a t e Univ., Columbia(0hio) 1969,

Authors' addresses

Dr. M. Ch. Klin N. L. Na jmark Staa:liches Pädagogisches I n s t i t u t Staatl i$hevUniversität "K. E. Ciolkovski jgt ''T. G. Sevcenkott

SU - 248023 Kaluga - Kiev

Dr. R. Pöschel Akademie der Wissenschaften der DDR I n s t i t u t fiir Mathematik

DDR - 1080 Ber l in Mohrenstr . 39