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 Z E N T R A L I N S T I T U T F U R M A T H E M A T I K U N D M E C H A N I K

M. H. Klin and R. Pösdiel

The König problem, the isomorphism problem

for cyclic graphs and the characterization

of Schur rings

Als Preprint gedrudrt

Berlin, März 1978

ABSTRACT

Connections between S-rings (method of I.Schur), graphs (in particular Cayley graphs) and their automorphimu groups (containing a regular subgroup) are shown in this Paper. As a concrete example of these c.onnections the cyclic @oup Z

PQ . (pfq prime numbers) will be considered for which the S-rings in the group ring Z(Z ) will be described; moreover, the

PQ König problem (characterization of automorphimu epups of graphs), and the isomorphism problem for cyclic graphs over

(i.e. Cayley graphs of Z ) will be solved. P9

Zpq is a -CI group, i.e., for isomorphic cyclic graphs over Z

PQ there exists an isomorphism belonging to && Z

~ 9 '

In der vorliegenden Note werden Zusammenhänge aufgezeigt, die zwischen S-Ringen (MethodB von I.Schur), Graphen (spe- ziell Cayley-Graphen) und deren Automorphismengzuppen (mit regulärer Untergruppe) bestehen. M e s wird am Beispiel der zyklischen G ~ p p e Z (pfq Primzahlen) demonstriert. Dabei

PQ werden die S-Ringe im Gmppenring Z(Z ) beschrieben, sowie

PQ das König-Problem (Charakterisienuig der Automorphismengnip- pen von Graphen) und das Isomorphieproblem für zyklische Graphen Uber Z (d.h., Cayley-Grophen von Z ) gelöst.

PQ PQ Zpq ist eine$-~~-Gru~~e, d.h., filr isomorphe zyklische Graphen über Z gibt es bereits einen Isomorphismus aus

PQ Aut Z - PQ'

PE-

B nacromeä paöore onacmu c ~ a a n ~ e x q ~ ~ S-xoxbqeun (ureroa II.üiypa), r p e a u n ( B uac~aocza r p e e u a Kaaa) E irx rpynnausi a~rouop@aauoa (co~epsaiquuaca peryaspnym no~rpynny). 3ra C B R B ~ no~aaellln Ha npnuepe I U ~ K ~ H Y ~ C K O ~ ~ rpynnn Z ( pfq P9 npocrue). Haitxewo onacezixe Bcex S-xoxeq B rpynnoBou Kanbqe Z(Z ). iionee roro, pemenn npoöneua K e n ~ r a (orüicanne P9 rpynii aaro~op@narno~ rpmos) a npo6aeua m a o ~ o ~ a ~ a B xaacoe qaxaaseoxax rp-B H= Zpq (r.e. rpa@oa Kann rpynnu Zpq) . Y c r a n o ~ x e ~ o , s r o Zpq aeaaezcn -CI-rpynnoB, r.e. M X a ~ y x 9 aaouopmxx qaKamsecKmc rpa#oa H- Zpq aymecrByer aaouop- @sau yxe n3 && z

~ 9 '

I n t r o d u c t i o n . Pe rmuta t ion g roups o f t e n a r i s e a s automor-

phism Eroups o f a l g e b r a i c o r c o m b i n a t o r i a l o b j e c t s ( l i k e

g r a p h s , hyperc raphs , r e l a t i o n n , u n i v e r s a l a l g e b r a s ) cha rac -

t e r i z i n g t h e " i n n e r symmetrytt o f t h e s e o b j e c t s . I n t h i s

paper we c o n s i d e r ( f i n i t e . d i r e c t e d ) g r a p h s snd t h e i r au to -

morphism groups. Two well-known problems a r i n e inmedia te ly :

A ) 'Ph~-Pgflig-pfc~bl_~~: C h a r a c t e r i z e a l l automorphisn

g roups o f grapt is (v~e c a l l them König g roups ) !

From t h e a b s t r a c t p o i n t o f vievr t h i s problem can be s o l v e d

e a s i l y [8]: Each group is isomorphic t o a König group.

But t h e c o n c r e t e c h a r a c t e r i z a t i o n a p p e a r s t o be a v e r y ha rd

problem and n e a r l y h o p e l e s s t o s o l v e i n t h e g e n e r a l case .

One h a s t o r e s t r i c t t h e c l a s s o f g raphs o r groups t o be

c o n s i d e r e d f o r o b t a i n i n g c o n c r e t e r e s i i l t s .

B) The isomorphism problem: Find " g ~ o d ' ~ n e c e s s a r y and - ----------------- - - - - - s u f f i c i e n t c o n d i t i o n s f o r g r a p h s t o be i sonorph ic !

Born t h e t h e o r e t i c p o i n t o f v iew t h e r e is n 0 t h i n ~ t o do:

There olways e n i s t s a f i n i t e ' a l g o r i t h m t o d e c i d e (by check ing

211 grnphs o f t h e naine c a r d i n a l i t y ) whether two ( f i n i t e ! )

g r a p h s a r e isomorpnic o r not . 3 u t even a high-speed computer

could not manage the l o t of computations mceeeary f o r t h i e

algorithm. Agaln, one hae t o r e e t r i c t the c laee of graphe

under coneideration f o r more effect ive reeulte.

Throughout t h i e paper we a re concerned mainly with graphe

(H~I), the vertex eet H of which i e endowed with a group

etructure such tha t the automorphiam group M@;&) con-

ta ine the r igh t regular repreeentation HA of H (euch I are

cal led Cayley graphe of H). Pepecially we are dealing with

cycl ic p u p e H = Zn (then the graphe i n question are cal led

t o be cyclic).

Were 1s a method of I. Schur ( lese known.than the method

of group charactere but able t o produce eimilarly deep re-

d t e although eubetantially more elementary) whlch i s

deeigned f o r the inveetigation of (2-cloeed) permutation

groupe G contalning a regular mbgroup HA of the eeme

degree. To every euch G correeponde a eo-called S-ring i n

the group r ing Z(H). I f convereely t o every S-ring corre-

eponde a penmrtation group GLHA(euch H ' s a re cal led Schur

groupe) then v ia the deecription of S-ringe we get a charac-

t e r i za t ion of a l l KMnig groups containing HA. With the me-

thod of S-ringe we can also t r e a t the ieomorphism problem

provided t h a t rill S-ringe a re non-ieomorphic.

In addition t o the applicatione t o graph theoret ic probleme

the r e d t e on S-ringe given i n t h i e note a re of t h e i r o m

intereet .

I n the f i r s t eection we give the basic def in l t ions , nota-

t ions and some resu l t e needed l a t e r .

I n the second sect ion we ou t l ine the rnethod of S-rings,

describe a l l S-rings over Zn where n 1s a product of two

d i s t i n c t primes (Theorem 2.8) and prove Zn t o be a Schur

group (Theorem 2.9). There 1s an easy f o m l a f o r the num-

ber of a l l 8-rings (Proposition 2.10).

I n the th i rd section we give a neceeeary and su f f i c l en t

condition f o r a group C t o be a König group provided tha t

C contains a Schur group a e a regular subgroup (Theorem 3.2).

Ueing t h i s r eeu l t a l l K6nig groups G containing ZA (pfq P9 primes) a r e determlned. They a r e exactly the 2-closed per-

mutation groupe which are given expl ic i te ly i n the Theorems

3.3 and 3.4. Thus even the automorphiem groups ( G z Z A ) of P9

coloured graphs are a l so K6nig groups. Theorem 3.8 shows

that a l l S-rings over Z a re paiEwlse non-isomorphic. P9

I n the fourth aection the isomorphism problem f o r Cayley 2 graphs of H ie considered (especia l ly f o r H=Zn, ncip,p ,pq)).

For groups H with pairwiee non-isomorphic S-rings we prove

i n Theorem 4.6 tha t there 1s a 1-1-correspondence between

the Cayley graphs of H ieomorphic t o a given Cayley graph

<H;&) and the elements of N (G)/G where N (G) i s the nor- SH %I

malizer of G=E(H;&> i n the Ful l symmetric group SH over H.

This ttieorem is the bas i s f o r the solution of the isomor-

phiem problem f o r cycl ic graphs, 1.e. Cayley graphe of Zn.

The r e s u l t e are mentioned f o r n = p, p2 (4.2, 4.3) and

proved f o r n = pq (pfq primes). Zpq turne out t o be a

9 - C I group. i . e . two Gnyle j grcrphs ;in6 8' o f Zpq a r e i s o -

morphic i f f t b e r e e x i s t s a n mt& Z such t l i a t & * = O" P9

( A l l ii z r e C I - ~ r a p h s , 'i'neorezn 4.7). Tlie pcrper ecdo w i t h

Sone r e m l t s On CI-yraphs ( i . e . Cayley ~ r z p h s o f 11 w!iich

heve t h e j u s t n e n t i o n e d p r o p e r t y w i t h H i n a t e a d o f Z P9)

Ttie &! o f t - i i s paper is t o show tlie c o n n e c t i o n between

S- r ings ( t h e method o f Schur ) , g r a p h s ( C q l e y ~ r ü p h s ) and

t h e i r auto!~orphisni Eroups ( c o n t n i n i n ~ a r c z u l a r subgroup) .

A s a c o n c r e t e exainple o f t h i s connec t ion t i ie c y c l i c group

Zpq w i l l be cons id9red f 'or which we s h a l l s o l v e t h e König

prob1e:n nnd t!.e Isoinorphisn problem f o r cy ' c l i c a r a p h s

ovcr 2 . bi Course , t h e s e p r o b l e x s a r e o f t!ieir otVtr. i n t e r e s t . T" J

i n p - r l i c u i : - r i 'or p r a c t i c z l a p p l i c a t i o n s .

i..ore abou t t h i s t r e ü t m e n t c a n Oe found in[23].

f d d i t i o n e l l y , we g i v e t h e f o l l o w i n g r e f e r e n c e s :

;,:etkiod o f Schur: [ 1 8 , 27, 333

I.önig problem: [ l l , 19 , 23, 29 , 311

Isomorphis!n p rob len : [1 - 7 , 1 Y, 2 3 1

Cayley g r sphs : [24, 25. 29 , 30 , 3 2 1

i'he end of a proof ( o r of a p r o p o s i t i o n n i t h weil-;:nor-tn

o r easy p r o o f ) 13 nzrked oy I .

ACKlCGV&iiüGL.iE;.T. ;ie wish t o e x p r e i s o u r tii~r.lr:: t o ;roi 'essor

L.A.Kalu%nin who had a i r e c r e d OUI- : : t t en t ion t o t i e i t ;vf ;? t i -

g a t i o n o f Schur r i n g s aiil shoaed z cof i t inuous i n t t x - e s t on

our-vtork. Our thafiks z r e a l s o du? t c i , .~;. t ir inj;u~kii? :'ehr

c o m e c t i n g a ni is take and f 'or h i c cooper i i t ion vt'l!.it. tt:o f i r s t

a u t t o r d u r i n g t h e p r e p a r ü t i o n o f t h i s ~ . z ; . e r .

1 ; Definition8 and preliminarr reeul ta

A l 1 se te and gronps are t o be f i n i t e .

1.1 We denote by % the f u l l ~yimo~mri_~ g m ~ p On a s e t H, - i n p*ticular Sn SI^ ,l „ , n-1) ( n r l = eet of natural num-

bere). Bvery vrSg i e a pernutation V: H + H : X c, xV . V V ' * For v,v8cSH the c2=~05i_44gg W' i e defined by xw :=(X )

For pernutation groups G acting on H , i.e. subgroups of Sg,

we use the notation

(G,H) or GfSH

( the eign L alwaye denotes the subetmcture property).

Por eubeete K , K V % and xcH we eet:

KK1:={wv I veK. v * t ~ * ) (and Kk:=KK* f o r K*={L)),

:=[xvI VEK).

G 4 % i e cal led tmggi_t&~g i f xG= H f o r xcH.

ITG'(G):=~gtG' I gG = Gg)

i s the norinaliger of G i n GIG%.

~ , r = { g t ~ I zg=s ) i e the gt_&&&4geg of C i % i n the point icH.

1 .2 Let (H; W, e) be a group with uni t element e. Then - 4 ~ u t H - the m t _ c = g ~ h i _ e g r g g ~ of H - i e a aubgroup of %. -

I The r&ght_ ::=&E cg~rgegg$g4iog (HA,H) of H i s the aubgroup

of % which coneists of the r igh t multiplications h' by

elements &H:

i ha:H--+H: x-xh (xtH, bH).

eA i e the ident i ty l

'Pbrou&out the paper we uee the e ign t o remember thaE we

have t o do wlth pemuta t ione (and not only with elemente of

a e e t H).

EI(H):= H (HA) s~

i e c a l l e d the h g ~ o ~ g h _ of H. We have

M(H) =(@ H) HA =HA (& H) M( L21 , p.3393 1.

Let; Zn = (z,;+,o) (2 4ncW) be the (addi t ive) cyc l i c

group on the s e t Zn= {0,l , ... ,n-1) (addi t ion modulo n with

Zero element. 0 ) and l e t P ( n ) =(lP(n); . , l ) be the (mult ipl i-

c a t i v e ) gmup o f a l l numbers a t z n r e l a t i v e l y prime t o n

(mul t ip l ica t ion modulo n with un i t . element; 1 ) . P ( n ) i e the

prime reeidue c l a se group and can be coneidered ae a pemu-

t a t l o n group ( P(n )^ ,Zn) , ehor t ly IP(n)^, ac t i ng on the

set: ZI, by r i g h t mul t ip l ica t ion:

an: Zn+ Zn: X o xa (xcZn, ae lP(n)) .

I n t he aequel, mu l t i p l i ca t ion and addi t ion i e alwaye taken

modulo n i f we a re dea l ing with Zn and we e h a l l uee the

nota t ion Zn f o r both the p u p and the underlying eet .

We have

IP(n)^ = &J Zn , therefom (cf . 1 .2)

H(!,) = IP(n)^Z: = Z n lP(nIA

coneia ts of all penrmtatione g = a A b A

g : zn -+ Zn : x ~ x a + b ( a r l P ( n ) , b o ~ ~ ) . ~ ) I X ) I n t h e aequel, it w l l l be c l e a r from the context o r men- t ioned se e r a t e l y whether X' i e an element of lP(n)^ o r ZG i f x c ~ ( n Y g zn.

For FsZn and mcZn we write P m : ={mn I xcF].

1.4 Lemma. 2 transitive s u b m u p s G fM(Zn) are of the form G = A'z~ where A^cIP(~)^. Yoreover, -

X, (G) = N (Z;) = M(Zn) .I n Sn

The ii_reci grfigc4 ( G ~ ~ G * * , x ~ s X ~ ) = (G' ,X9)x(G**,X~*)

of tvm permutation groups consists of all pemutations

(g'*g"):X'iXtL,x*xXii:(x,y) » (xgi *yg*i).

The zgrlb grofi!c$ (GqzGn,X1.X8*) = (Gl,X*)2(GM,X*1)

1s the set of a11 pemutations of the fonn

V (U *-f

:X'.XW 'X*.X":(x*y) W (XUIY&(~)),

where ucG1 and d : X' + G" 1s an arbitrary mapping. FOr we write also (U*(W~)~,~,) where wx:=o<(x)~Gn.

1.6 Lemma. E transitive permutation groups G'LSXl - and GnCSX„ following holds (X:=X1rXn): -

( G ~ L G ~ * ) =I (U, ( w ~ ) ~ ~ ~ ~ J U ~ N (G, 1,

wx6N (Gw) and wxlrwxG~~ SX'l

for all X , X ~ ~ X ~ 3 . We neglect the proof of this group theoretic result, but it

1s easy to See that the right side ie contained in the left

one. For the other direction one has to do a little bit

more. I

1.7 We need eome notatione f o r dealing with the additive

cyclic group Zn where n = p q i e the pmduct of tim primee

p b q . %er8 exle t two numbere (reeidue claeeee) e and e P 9

auch tha t

eP a 1 mod q , eq s 1 mod p , e r O m o d p , e a O m o d q , P 9

Every acZ, has a uaique repreeentation

a = alep + ane (altZq, anrZp), 9

where a l s a mod q, a n i a mod p.

The mappinge a » (a ' ,aW) and a +P (an ,a l ) areisomorphiame

from the group Zn onto Zqap:= ZqxZp and Zpaq:= Zp*Zq, reep.,

(862,). Moreover, they a re ieomorphisme from P(n) onto

P ( q ) r ~ ( p ) and P ( p ) i P(Q), r e e ~ e c t i v e l ~ , ( a r P(n)) . l ( [lol)

Throughout the paper we f i x ep and e ( f o r given p and q) 9

and make no differente between Z n = Z p q , Z q e p and Z P'9

( 8 1 . 9 ~ 8 ueing the above ieomorphiame).

Thue the pennutatione (gl,gn)cSqrSp can be considered a s

act ing on Zn by

(g9,gn)8 alep + a e +w a1g1e + a2gne 2 q P 9

(alcZq, a2eZp) and (O1,Zq)x(C",Zp) l1isn a permutation group

act ing on Zn.

Por W 1 5 Z q and W1'CZ we v r i t e P

wge + wneq:=lw'e +wne J WCW'. WYW~]. P P P

1.8 Lemma. n = p q (pbq prime numbere) we have -

The p m o g follows from 1.3 and 1.7: IP(n)^Z2 IP(q)^ZA P(p)^Zi 9

1.

1.9 Lemma. U: (Gt,Zq) (Gn,Zp), resp., be e i t h e r tran- . - e i t i v e eubnrou~s of the holomomh ( ~ ( 2 ~ ) o ~ : H ( z ~ . ) , - 0 )

1 o r the f u l l symmetric p u r , . Then

1

7- m-og. Eecause of 1.4 and 1.8 one hae to consider oniy the

case where a t l eae t one of the G' o r GI8 1s the symmetric

group. We l e f t t h i e case t o the reader ae a group theoret ic

exercise. I

1.10 ' For a givan group H=@;- ,e ) the g p g p riog (z(H);+,.)

(over the r ing of integer8 z ={... ,-2,-1 ,O ,I ,2, ...) coneiete

with addition and muitiplication defined a s follows:

X ahh + X bhh := (ah+bh)h

(z(H);+) 1s also a Z-moduie with sca la r muitiplicatione

defined by ' ?

The guggcggl_g gg Z(H) ggfiggatgg (spanned) gg elemente

?oZO„,Tn€Z(H) conaists of all l i n e a r combinations n

ciri C Z(H), (ciez>.

For f f H we define 1 i f hcT,

- h€H

and c a i l a l l euch (TaH) gLmp;e gum41,4&gg of Z(H).

1.11 A binary re la t ion iigHxH On a 8et.H i s eaid t o be

i_nvwgt_ f o r a pemutation vtSH (or V pmggrvgs a) i f

(x,y)rii impliea f o r a l l x,yrH. For a s e t Q of bi-

nary re la t ions the s e t of a l l pennutations preserving each

ircQ - denoted by & Q (or *Q o r &{H;Q)) - f o m s a sub-

group of % - the gxzt_cmnlpgi_n gggup gg Q. For Q 451 we

n r i t e fi iö instead of Q. The s e t of a l l binary re la-

t ions invariant f o r each vCG SH we denote by loZ(2)~.

GC2' ,= Aut I ~ V ( ~ ) G

i s cal led the = c l g - e e of G and G i S H is cal led - c l o s e d

i f Gt2'= G. ( 123. 341 )

1~.12 Lemma. p e m t a t i o n g r o u ~ s (G',Xt) & (Gn,Xn) - the following holde: - ( i ) ( ( G ~ * o ~ ) ~ ~ I , x # ~ x ~ ) = ( G # L ~ ~ . x # ) = ( G ~ ~ , x ~ ) ,

( i i ) ( ( G ~ L G ~ ) ~ , x # ~ x ~ ) ~ = ( G ~ ~ ~ ~ , x ~ ) z ( G ~ M , x ~ ~ ) . m( 11 21).

1.13 A (directed) e g p & ( ~ ; 6 ) - o r shortly 6 - is a bi-

nary re la t ion 8gHxH on the oegigx p q H. The elemente

(x,y)ea are cal led d g g g wlth the g g ~ g vertex X aud the

y ( and wl l l be represented by arrows from X t o y).

WO graphs (H; L) and (E; L') a re ig~n~n~hhg i f f there 1s a

v&SH such tha t (x,ykL +-> ( x V , Y V ) ~ ~ l f o r all x,ycH.

A family

a = (5, „,ar)

of pa imise d i s jo in t ( I ) graphs on H 1s cal led a

g d g g m b grgph <H;L) where (x,y)eLi a re the edges wlth

colour i, (it{l , ... ,r)).

-L := e { ~ ~ l i = i ,,,r)

1s the automorphiem group of the coloured graph L. We have:

G 6 3l & 2-closed iff G 1s the automorphism g-ig 2 coloured Rraph. I ( t ake the o r b i t s of G on HrH).

1.14 Let. (H; ,e) be a group., A graph (H;%) 1s cal led t o be - a Cax4ex g r a ~ h no H if

H A 5 & t H L .

( ~ ; o ) is a CE=grgph H (CI = gayley 'somorphism property,

[3]) if a Cayley graph (H;L')of H 1s isomorphic t o L i f f

there ex i s t s an mcA& H wlth 1' = (={(P,$) I ( X , ~ ) ~ L ) ) .

H 1s cal led a czrp-m~ grgphh, shortly a - C I group, 8 - - - - -- i f all Cayley graphs of H are CI-graphs.

1.15 A Cayley graph L of Zn 1s cal led a cxgdi_c (circulaut.) j

I grgph p g r Zn . Clearly, L 1s cycl ic over Zn iff (x,y)er + (x+l,y+l)rL ( for a l l x,y6Zn).

, There 1s a l-l-correspondence between the cyclic graphe

a over Zn and the subsets of Zn, namely:

a » r := 4 ~cz, 1 (0,~)ti6 f , r e a :={(X,~)~H=H 1 y - x ~ ~ ) .

Thus, throughout the paper ne shall speak of the cyclic

graph F E Z n instead of the corresponding L and we use ths

not ation

AUTF:=Aut,L ,(l'=Zn). 7

n

A co&eded cyc&ic grr~h 1s a coloured graph L = (L1„.9Lr)

the components iii of which are cyclic graphs (over Zn).

From the definitions and 1.3 follows immediately that

Zn 2 9 -CI group lf (and only if) two cyclic maphs

i?,F 'sZn isomorphic iff there exists mclP(n) (pA& z,) such that I?' = P m . I --

1.1 6 Example. The cyclic graphs i? ={I ,2,9') and I? *=[3,4,7)

over Z10 are represented in the following figure:

F I' '

'&e method of S-rings (Schur rings) can be used f o r the

investigation of groups containing a regular eubgroup ( f o r

more d e t a i l s see [18, 331). Roughly speaking the S-ring of

a permutation group (G,H) is - i n some sense - nothing e lee

than the r i n d o f a i l binary re ia t ions over ii invariant f o r G

f wlth reepect t o the re la t ionai product).

2.1 Definition. Let H =(H; ,e) be a (nontrivial) group. - A r ing SIZ(H) 1s called an S=gi_ng p g c H i f the followlng

holds (cf. 1.10):

(1) S i e a subring of the group r ing (z(H);+,=>;

( i i ) S 1s generated (as a submodule of the Z-module Z(H)!)

by simple quantit ies To, Tl „, T, (noB) euch that. - - - n n

( I ) T e T = i . . Ti= H dle joint M o n ) and = - 1=1

(h" -inverse (iv) r = & a h h h h s + +(-l):=& ahhdeS. of h i n H)+

'Nie To, Tl , , T, - o r the s e t s To, Tl , , , T, - are .called - - - the +g&c ggan$14&gg of the S-ring S. A @ k S = f i n e ; of S 1s a

S-ring S' over H the basic quantit iee T;,Ti, .-, T;, of d i c h

are sums ( in Z(H)) of some basic quant i t ies of S (notation:

S' sS) .For an S-ring S we denote bg

o r ahortly T(x),

the uniquely d e t e d n e d ( 1 ) baeic quantity T(x) auch tha t

xeT(,) . The S-ring generated by the tao basic quanti.tiee

~ ~ = { e ) and Tl = Mle} is called the ~E&G& %eog.

-1 4- . 2.2 Definitio~. L 8 t . H be a group. Por a permutation group - (G,H) containing the regular eubgroup (HA,H) let ! ( G + ) be

the Z-eubmodule of Z(H) generated by the orbite of G,:

I(G,~) :={G ch% I CheZ, T(h)= 2'. hdi].

1.Schur[27] found the following result r

The conjecture of Schur that each S-ring over H 1s of the

f o m ~(G,H) turned out to be false ( [33, yheorem 26.47, [221).

Thus we define:

2.4 Definition. An S-ring S over H ie said to be ~J-IG!E~~ . if there exiete a penrmtation group (G&) containing (HA,H)

euch that S = i(G,H). A group H 1s called a if

ail S-rings over H are induced S-rings.

We will define epecial ieomorphisme between S-rings (over

a group H) eich are given by wunderlyingw permutations of H:

Definition. A 1-1 -mapping 1: I1 j H E % ie cailed an S-ieomoqhism of the S-ring S onto the S-ring Sg(over a group -------- ---- H) if

(T?„~P = TS' h2 ~ O P aii 3,hm; (E")

in particular,we have 8% = e (eet z=e),.

Then S and Sg are called to be S=i_ggmonogh&g. (The producte

Th are meant elementwise(in H) : Th ={th 1 t ~ ~ j . The eubgroup ( 1 )

AUT s i i { ~ t ~ H 1 VX.Y ,ZEH: y ~ ~ q e ) x + & q,,2} - of SH i a cal led the group of all (quag-1 S=gg42~2ghidg

of S.

Remarks :

( i ) Obviously, we have Has= S and

AUT S = (E SIe Ha = R ( E SIe -- where ( E S ) , i a exactly the s e t of all euch S-isomorphitnns

of S in to i t s e l f which preserve a l l basic quantities.

( i i ) Fron the definit iona follows:

AUT S = f\ Aut 6 - erH - T ( ~ )

where 6 ~ ( z )

:={(X~Y).H.H I yx-'C

i i i i ) Moreover, f o r each simple quantity KSH 'and

~ ~ : = { ( x , y ) I ytKx) we have

AUT S &H BK g € S f o r each S-ring S, - i n par t icular ,

CSAut„oK 93 K€S(C,H) f o r a l l groups (C,H) L(Ha ,H).

2.6.Theorem([23, 3411. - (Al 2 : (C,A) »S(C,Y), (CzH"), and A U T : S c * E S

define Calois correspondence such that -- ( I ) G = U S ( G ; H ) W C W = ~ ( ~ I i " ) ;

( i i ) S = g ( E S , H ) S i a a n i n d u c e d s - r i n g .

(B) H 9 a Schur m u p . i f f AUT S # S1 f o r all S-rings

S # S' - B . 1([23])

Remark: Fmm 2.6 - o r d i r e c t l y from the de f in i t ions - ioi iows G[']= E s(c,ii) and I (~b ' .~ ) = i t ~ . i i ) i o r iii ~ a k .

Now we are going t o detennine exp l i c i t e ly all S-ringe

over the cycl ic group H = Z n where n - p q (pfq prime numbere)

b r meane of olS-eystemsn. Por n=pe(seBJ) see , e.g.,[22.23].

Remark t h a t from now on H i s m i t t e n additively: 'I!Q avoid

confusions we s h a l l not (and need not) use addit ion and

eca la r mul t ip l ica t ion ( i n Z(H)) of the S-rings such t h a t a l l

aume and products w i l l be i n Zn(i.e. modulo n ) , i n part icu-

lar we have: T+x:={t+x I t c ~ ) , !P+T':={t+tl 1 tLT, t h ~ * J and

T X r = j t x I t c ~ ) f o r xczn, T,T*S zn.

Definition. Let 2inCB: and l e t S be an S-ring over Z„

D, = {do,dl ,- ,dk-ll the s e t of a l l d iv isors (f n) of n,

d O = l , and f o r KCZn and deDn define

K/d := { xrK ( g.c.d. (x,n) = d).

The binary r e l a t i o n B(S) on D, defined by

(d .a l )c e t s ) :(P) T ? ~ ) / ~ I f B , ( d , d l ~ ~ n ) .

is an equivalence r e l a t ion ca l led the btxjic ~ g ~ i ~ & L g g g g gg S.

Por the aubgroups Ad(S), shor t ly Ad, of ( B(n)^.Zn) defined

b~ Ad = ld(s):={x6lP(n) I = , (dcDn),

the sequence

i s c a l l e d the Szgygtem ($X:: IP(n)) of the S-ring S.

semarks: ( i ) (d,dl)€e(S) + Aa(S) = Adi(S) ; - ( i i ) s'os 19 e(S)C 8(S9) and Vdc~,: A ~ ( s ) c A ~ ( S ' ) .

For n -pq (pfq prime) there are 5 equivalences On ,

= { I ,p,q\:

8, ={(I ,1 ) , (p ,p) , (q ,q)f , Q1 = D n r D n ,

ep = e o u {(I .p).(p*l I} 9 Bq = Oo U {(I *q)*(q*l )I. epsq = e0 U t (p ,q ) , (q ,~ )} .

In t h i s case, the S-systems w i l l d e t e d n e uniquely the

I J S-rings a s the following theorem shows.

2.8 Theorem.

(1) nontr ivia l S-ring S over Zn, n=pq (pbq grime) ,&

I an S-srstem g(S) = (A, ,$,A ; 8 ) of one of the following fonns - 9 ---- 1 (notations sec 1.7):

a ) Ap = A, ep + IP(p)eq, Aq = P(q)e + A,eq , A, 5 IP(n), P 8 = 0 0 ;

b) A, = % = W 'ep + IP(p)eq , A = IP(q)e + Wweq , 9 P

e = e W'S P ( q ) * W"dIP(p); P * I C ) Ap =W9ep + IP(p)eqs A, = Aq = P(q)e + W1'e

P 9 ' e = eq. W'flP(q), W"SP(P).

I . 5 (2) E v e q S-srsteio of the form (1 )a),b) ,C) i~ the S-system

of e x a c t l ~ one nontr ivia l S-ring S & Z where S -- - Pq - rated dy the following basic auant i t ies : T(o) ={o) ,

,O a) T(z)= A,z, T(pz)= A,Pz, T(qz)= A,QZS ( z t P ( n ) ) ;

b) T(,)= B'z+Q' (= T (pz, ) for PZ ' s z mod q ),

1 T ( ~ ~ ) = wl'qz , ( z e P ( n ) , ~ ~ : = { 0 , q , 2 q , - ,(p-l )q) 1;

C ) T(,)= I "z + Q" ( = T f o r qz 3 z mod p ) , (qz) -

T(pz)= :'ilpz, (ze lP(n) , ~":={0,~,2p,., ,(q-l 1.

P ~ C I X I ~ . It i s easy t o see t h a t the S-rings given i n (2)

define the S-systems i n (1) . Yor the opposite d i r e c t i o n one

can show tha t d i f f e r e n t S-rings have d i f f e r e n t S-systerns.

It remains t o prove tha t a l l possible S-systems a re des-

cribed by (1 ) . This can be done using some techniques f o r

S-rings (given f o r instance in[33]). For more d e t a i l s

see L23. iiauptsatz 3.4.1d-j. I

The fo l lo s ing theorem was mentioned but not proved i n [23].

2.9 Theorem. Z (pfq prime numbers) a Schur aroup. - P q

o m c f . We want t o use 2.6(B). For an S-ring S over Zn(n=pq)

vie ge t (E S ) o n P ( n ) a = A i n 5 n AL

(since hcAd => ( y - x ~ T ( ~ ) 3 y h - x h ~ S ( ~ ) h = T ( ~ ) )

=> hae(= S)O ( s t a b i l l z e r of the zero element 0 ) ) .

' I h e r e f o ~ , frorn 2.8(2) we obta in I\UT S f S' i f the

S-rings S f S 1 are of the Same type a ) , b ) o r C) . On the

o ther hand, f o r an S-ring S of type b) the mapplng

3 : Zn 4 Zn def lned by

f o r XE?' (see 2.8(2b)), + otherwise,

can e a s i l y proved to be ans-aitomorphism of (E SI0.

Obviously, ? S' f o r a l l S f of type a ) o r C) . Analo-

geously, f o r every S of type C ) there e x l s t s a Xe(= SI0

-1 9-

with A #E S' f o r a l l S-rings S1 of type a ) o r b) . Thus,

AUT S # S' (E S ) O # ( a OC) S f S9 and t he - proof is f in ished by 2.6(B). I

As a co ro l l a ry t o 2.8 we obtain:

Proposition. There a re

Sn U n + 2U U + 1

P 9

S-rinas over zn (n=pq, p#q primea), where uk denotes the

number of a l l e u b m u p s P ( k ) , ks{n,p,q). P--

2.11 Remarks. Prom 2.8 and the de f in i t i ons (2.2, 2.7) we - get: ( i ) E S nH(Z,) ;. A, (S)^Zi f o r a l l S-rings S

over Zn.

( i i ) A, (S) = ~t~ ) and S = E Al iS ) (4 considered ae

a cyc l i c graph, cf. 1.15) f o r a l l S-ringe S of type 2 .8a l . l

We w l l l f i n i e h t h i e sec t ion with a concrete

2.1 2 Example. - Let n - f 0 = 2 * 5 , p = 2 , q = 5 , e = 6 , e q a 5 . We have:

P

[ b ~ ; ! ; ; g l p ( n ) r ( q ) , P ( p )

f W; =(1) Ti; ={I\

W; =I? r4)

w2 * 3 , 7 , 9 ) = ~ ( n ) ={I ,2,3,4] i

From theorem 2.8 we ge t the following list of a l l S-rings S

over Zn. We mention here a l so the correeponding automorphism I w u p s which w i l l be de temined i n 3.3.

3 . The Könin problern

We consider the König problem f o r cyc l i c graphs, i .e.

t he concrete charac ter iza t ion of automorphiam groups of

graphs which conta in a c y c l i c group Zn(n6N).

1 3.1 Definit ion. A pemuta t ion group CG,H) i s c a l l e d a J -.- iigig p - g p if the re e s i s t e a graph (H; ii) with

G =E„&. ..

With the following theorem, the c r u c i a l point f o r t he solu-

t i o n of t he König problem tu rns au t t o be t he deecr ip t ion

of a l l S-rings.

3.2 Theorem. =.H =(H; ,e) & IA Schur mup(cf .2 .4)

(o r an a r b i t r a r y p u p . resp.) (G,H) 2 2-closed permu-

t a t i o n containinp, the r e m l a r s u b m u p (H" ,H). - 'hen

G 2 a König m u p i f f the re e x i s t e a simple g u a n t i t g 5 i n S = S(G,H) such t h a t K f SI f o r a l l proper (or a l l induced - proper, resp.) sub-S-rings S' S. I n o the r words, G &a=

König -ur, iff S(G,H) Q generated ae- S-ring b~ a a i w l e

I simple q u a n t i t r (provided H a Schur m u p ) .

m g . For G==„& def ine K : = { ~ ~ H I ( e , h ) € ~ t . Then

7 Ke S' ++ G I = S ' M S(G,H)s z(E S',H) =S' (cf. 2.4, - - 2.5, 2.6; S' i a induced!) and therefomK i e a quant i ty of

no sub-S-ring S' of S. On the o the r hand, if eome g has ?

The f igure given here showe the q l a t t i c e (with reepect t o inclu- eion) of the S-rings S over ZlO.

S' -1 SN -1 Thie l a t t i c e i e antiieomorphic t o the l a t t i c e of a l l 2-closed eub-

9' -2 1 groups (=E SI of si0 a c h

cont ein ZTO.

- AUT S

zi V'-2- 1 q xSp

S q s S P

2; 2 SP W,'-Z; t S p

Sq S~

SP 2 2; sp 2 W;-2;

S~ Sq

A., (SI=

wo W,

w2

W 1 =

W6

W i

(W1'=WC;)

V'=

B W;

(W"=W") 0

1 9-ring 9 eet of a i l baeic quantit iee

of S

type a ) ( 9 = 9 0 ) : --------- -------------------------------------..------------ so I fo,~.n.z.e, 2.4.6.8 ,2 51 f0.19.r;1.3g.e6,Pi

S -2 10 1 a3.7~9 i 294.6.8 2 1 --------- -------------------------------------..------------ type b ) ( 9 = 9 ): --------- --Q----------------------------------..------------ % {oilLs,z~e.L2*ss#r) S' -1 {0 1.63.4 e3.8.7-2 941 S' -2 o , I .2.3.4.6.7.8.9 , ~ i 1- --------- ------------------------------------.-------------

type c ) ( 8 = 9 ) r ---------- -9----------------------------------.------------- % {0 1.3.5.7.9 2,4,6,2 f Si' - 40, 1.3.5.7.9 . 3 g , u j

25 { 0 1 *3*5*7*2 I , 2.4.6.8 )

i-------- ---------------------------4+-------..------------

tl-ivial (8 =el 1: L--------„--------------------------„-------..----,-------

%o {Q , 1.2.3.4.5.6.7.8.93

t M s property then we get f o r G':=&& (cf. remark ( i i i )

t o 2.5): G ' 3 G + KQS(G',H) f S(G,H) =+ P(G',H) = g(G,H)

+ G a = G*U1= GCzJ= G (cf. 2.6). .(cf.[23, Satz 8.5.51).

Remark. In general, a 2-closed group need not t o be a

Kt)nie group (see l23, 8.5.63). But f o r pemutation groupe

containing Z; t h i s property 1s f u l l f i l l e d i f f o r instance

ne{pq,p,p21. The following theorem ahows t h i s f o r n=pq:

l & & - t _ h ~ ~ ~ ~ ~ (Characterlza*ion of the automorphlern

groups of cycl ic graphs over Z pq)'

Let G f Spq(p,q W, p+q) iZ~qlZpq)L(G,Zpq). Then the - followirip: conditions are epUiy%&&:

(1) G& 2-cloaed: G'~'= G;

(U) a Kt )n i~ m~. i.e.,3 s t zP G = E ~ i ; P9

( i i i ) ff has one of the followinq forms:

a ) G = Spq ;

b) G I #I, B 2 g t r a n s i t i v e a u b m u . of the Li Li holomoruh H(z ), (&, E!=A Zpq f o r so-

P9 s u b m u u An f lP(pq)-, cf. 1.4);

c l ) (G,Zq.p) = (GIZGw,Z rZ 1, 9 P

c2) !C,zp*q) = (GWZG1,Zp~Zq),

where (G1 ,Zq) g& (Gtg,Zp) a r e König groups on Z and 9 -

Z containing Zn arid Z p , resjpectivelx. P 9 - ,-

(Use Zq.p=Zpq=Zp. q, cf. 1 '7).

Before proving t h i s theorem we want t o give a more e x p l i c i t

characterization of the p u p e G described i n (1i i )b) ,cl ),c2>

3.4 Supplesentary theorem.

(1 ) & 2-cioeures G = aL2' G t r ans i t ive s u b m u ~ s B

N(Z ) 3 exactlg following moups: P9

b,) G=G'rG1@ *), where (G9,Z ) & (GBB,Z ) are a s i n 9 P - -

3.3( i i i )c) ;

b2) G = AAZA where A^ g eubarou~ IP (pq) bu t not P9 -

a d i rec t product (1.e. AfW1ep+Wfteq for Wg5IP(q), - -- WMCIP(p)).

(2) The K6nig m u p a (C , Z ) (p prime) containiw, (Zi,Zp) P

a re exactly the following p u p s : - d ) C = S

P '

ß) C = A"Zn where AA 9 a proper s u b m u p G IP(plA. P -

Yoreover, G 2-closed p u p s 2 Zn a re König groups. P-

Eogg 2-2. (11) =+ ( I ) t r i v i a l .

( i ) + ( i i ) : By theorem 3.2 (and 2.9) we have only t o show

tha t every S-ring S=S(G,Z ) contains a simple quantity 5 P 9

such tha t 5/s1 f o r a l l proper sub-S-rings Sv of .S. Let S be

an S-ring over Z and C,(S)=(A, , A ~ , A ~ ; 8),(cf. 2.8). P 9

Cgg2.-1: S 1s of type 2.8a). Then 5 1s contained nei ther in

any S-ring S1 of type b) o r C ) (obviously, since ~ ( s ) @ ( s * ) ,

cf. 2.7) nor i n a proper sub-S-ring S' of type a ) because

of Al (S) $ A, ( S v ) (cf. rernark ( i i ) t o 2.7). Set K:= A, . *) fiiore i n de ta i l : G=G1e +GBBe consis ts of a l l (gl,gnl)rCsG*

P 9

i; a c t i m on Z a s described i n 1.7. P9

Gmi-2: S is of type 2.8b). Each non t r iv i a l sub-S-ring S'

of S mst be of the same type (since Q(S)S Q(S1) , remark ( i i )

t o 2.7). K:=(Ug+Q') V WpJe (see 2.8b) ) belongs t o rio

such S'.

hge-2: S 1s of type 2 . 8 ~ ) . Analogeouely a s case 2.

( i i i )=> ( i ) : It remains t o show t h a t t h e groups of t he

fo rn cl ) ,c2) a r e 2-closed. By 1 .I 2 ( i i ) we get

G*[212Gllklt G ' z G l t = G.

( i ) e ( i i i ) : I n ( i i i ) a ) , b ) , c l ) and c2) there a r e described

1 , upq, uqup and U u (cf. 2.10. 3.4(2)), respect ive ly , P 9

p e m t a t i o n groupe. It is easy to. see t h a t a l l these groups

a re d i f f e ren t and - by the above proof - 2-cl?sed. Thus we

have u +2u U +1 d i f f e ren t 2-closed groups ( G , Z ) contai- P9 P 9 PP

ning 2' Because of 2.10 and 2.9 (cf. 2.4, 2.6) the groups ~ 9 '

given i n ( i i i ) a r e exact ly a l l 2-closed groups.

This f i n i s h e s the proof. D

oEoof oC l h e s~~~l_eroentary theorem 14. We s h a l l not prove the second pa r t (2) which follovis from

a theorem of. ~ . ~ a l o m a [26]. For more d e t a i l s see [23; 8.5.1 51.

2; (1 ) : Let n = pq. I f A^S IP(nIa s p l i t s i n t o a d i r e c t

product A=W1e +\'(l8e (.;i1:&IP(q), Wa8tlP(p)) t!en a l so B=AaZi P 9

s p l i t s : B = B'nB", B* : = W r ^Zn B1*:=W1faZa Thus we have P' P' .-- ----

~ ~ ~ ~ = B ' ~ ~ ~ ~ B ~ ~ ~ ~ b e c ~ u s e of 1 .12( i ) where ~'Wand ~ " l ~ l a r e ju s t

t he wanted'König groups a s described i n 3.4(2) end t h e . .

case '(1 )bl ) 1s exhausted. It remains t o prove tha t B=P^ZK

4s 2-closed i f A does not s p l i t a s above (what exhaust the

caee (1 )b2) :

Considering the elements of ACIP(n) modulo q and p, resp.,

we define Av:n A mod q, An:= A mod p, Bv :=AtaZa gS , 9 9 B" :=AvvaZpC Sp. Clearly , B S can be coneidered ae a eub-

9P direct pmduct of B t and BIv; thue B @ ~ ~ ( B ~ = B ~ ~ ) ~ L B ~ [ ~ ~ ~ B ~ ~ I i e a eet of paire g=(gv,gn)(act ing on Zn, c f . 1.7) with

g ~ 6 B ~ f 2 J , g~EBnkl.

I B i a U o a ( z d , 1.0. B ~ ~ ~ Z Z ~ (ÄcP(n)) ,

then B ,= B" because S = g(B,~n)=g(~[211~n) a Sv (cf. 2.6)

h p l i e e A = ~7~ )= T:; )= Ä (cf. 2.2, 2.11 ).

kJ h n n e now B & B ~ then a nonlinear h=(g;.*)~B \Ei(zn)

muet ex i s t and a t l eae t one of the componente, f o r inetance

g;, muet be nonlinear, i.e. g ; 4 ~ ( ~ q ) = P ( q ) a ~ i , ( l . 3 ) .

Since HS (Za)=H(Z )(cf. 1.4) and eince za i e the only cyc- 9 9 9

l i c aubgmup of H(Z ) of o d e r q there exiete an element 9

4 ' 6 ~ mch tha t g;:= g;-lh; g; i e aleo nonlinear.

h f i n e G I : = i g 9 c ~ q 1 (gg ,U) r ~ l ~ l ) ( ~ = ident i ty on zp).

G v containe Z i and i e 2-cloeed (becauee g = ( g * , g " ) & ~ [ ~ ~ and

gtq= - i d i f f g ~ ~ [ 2 J and g preeerves a l l unary relatione

containe the nonlinear permutation g;. But, by 3.4(2),

a 2-cloeed penuutation group Gv6Sq containing a nonlinear

g,' hae t o be S Thue ~ [ ~ ' s p l i t e in to the direct pmduct 9.

G I S GI# 01 ~ a n i g groupe ( h e r e ~ ~ ~ : = j ~ * t e ~ ~ I (y,gll)e~[2J). B U ~

An= BOnP(n)^= Z n = *I (g(3L2',~n)) = (2.2.2-7) (2.6)

contradicts t o our assumption that Aa does not e p l i t in to a

direct product. Therefoniour asaumption B L J2]is f a l s e , i.e.

B 1s 2-closed. I

Corollary. For evew cycl ic coloured gp&

6 = (61,...,6r) Zn (n=pq, p fq primes) exiets c i cyclic

g-Eh 6 ' over Zn with the same automorphim -:

Aut 6 9 AUt 6'. , z -Zn n

mag. G== 6 1s 2-closed (1.1 3) and therefore a König

gmup by 3.3. I

Th18 w i l l be demonstrated by the following

).6 Bramples. a ) n = 10 = 2.5 (cf.2.13). For the cycl ic

graphs r1 =[2,8\ , P2 4 5 ) and I! ' ={I ,9 J we have

AuT(rl,r2) = r 1 (= u r 2 ) ) . - b) n = 15 = 3.5. Por the cyclic graphs r1 =[3,12), T2 =[6,9),

I!3 45,10) and P * ={I ,4,11,14) we have

AUT(l'1,1?2,~3) = P ' . - Remark. I f one hae a l i a t of (the basic quantit ies of)

all 5-rings S over Zn and t h e i r automorphism gmups & S

then i t 1s easy t o determine the automorphim eroup f o r

every given cyclic graph ' P S Zn by the following algorithm:

Look f o r the l eas t (with respect t o inclusion) S-ring S

auch tha t P 1s the union of basic quantit iee of S. Tlien

-27-

AUT F = E S. I ( t h i s follows from the remarks t o 2.5). - The description of a l l 2-closed.permutation groups, 1.e.

the automorphism e;roups of a l l S-rings over Z i n theorem ~ 9 ' .

3.3 fac i l i t a t es the proof of the following

Theorem. The S-r iws over Z (pfq primes) P9

pairwise non-S-isomorphic.

W-mg. Let 1: Zn + Zn (n=pq) be an S-isomorphism bet-

ween tvm nontr ivia l S-rings S and S' (cf. 2.5). We are going

t o prove S = S 8 .

Comparing the ca rd ina l l t i e s of ~ 7 ~ ) and T?:) (=Zn) we get

tha t S and S' must be of the same type 2.8(2)a) o r b) o r C) ,

and f o r the cases b) o r C) they must be equal (S and S' are

uniquely detennlned by t h e i r S-systemsl see 2.8).

Let S and S' be of type 2.8a). A d i rec t ver i f icat ion ahows

'' (zeB(q) ), respectively, that ~ 7 ~ ~ ~ ) (zelP(q)) and T(zep)

are the basic quant i t ies of S-rings S(p) and S(p) , resp.,

over Z e (equivalently, we could consider these quant i t ies 9 P

modulo q t o obtain S-rings over Z ), and tha t A res t r i c ted 4

t o Z e gives an S-isomorphlsm 1' of S(p) onto S(p). 9 P

uuiiogeousiy, the ~ t ~ ~ ) l ~ l d ~ 7 : ~ ~ ) (zcB(P)), resp., gene-

r a t e S-rings S and Siq) , resp., over Z e which are (9 P 9

S-isomorphic under 1" - the r e s t r i c t i o n of 2 t o Z e - P 9'

By the remarks below (3.9),we have S(p)=S{p) and S(q)=S{q),

consequently A, ep=qep and A, e q = q e q (for A, :=A, (S) ,

+::=AI(S1). cf. 2.7, 2.8a)).

Bow, f i r s t l y , we show tha t 1 s p l i t s in to ';X1 : Becauee

h 1s an S-isomorphiem (2.5) we have

%' = A., (zep) +(ze I" mä. analogeousiy. X

9 6 (zep+A, zeq)'= (zep)' + A., (zeq)' . Therefore

Secondly, we show tha t F:== S and F':== S v are equai.

From the definit ione (2.5) follows F'=' 1-I F 2 .

Looking a t the posaible automorphiam groups F and F' given

i n 3.3, 3.4, we have t o dietinguish tm casea:

a ) Both, F and F', a re of the form 3.4(lbl):

F=G'rGw (=G'e +Gnle 1, F'=(x' ~~~')~(~-'C~>~'),(~splitsl). P 9

If G'= S (caee 3.4(2&)) then obviously G ' = X"G 1' . I f 9

C1=AaZi (Aa(P(q)", caee 3.4(28)) then the S-ring S(p)=S{p)

1s nontr ivia l and the S-isomorphiam A' of S ( in to i t s e l f ) (P

1s an element of JP(q)*(see remarks below, 3-91, thus

G ~ = ~ ' ~ G ~ ~ ' . Analageously, we get the eame r e s u l t f o r G'* and

2". Therefore F'= >-'F A = F . b) Both, F and F', are of the form 3.4(lb2):

F=AaZn, F'=A'"Zi,(A, A' 4 P ( n ) ).

Then F'= 1 - l ~ ~ impliee 2; = A - ~ z ~ A , i.e.XrH(Z,);

therefore we obtain F = h-I F 3 = F' (cf .l.4, 1.3).

Thia f in i shes the proof becauee

S=S(F,Z,) =%(F1,Z,) = S n , (cf. 2.6). 1

Remarks. The properties of S-rings over Zp (p prlme)

needed i n the above proof a r e the followhg:

Every nontr ivia l S-ring S over Zp 1s generated by baeic

quantit iee of the form

q z ) = rs 9 (z.Zp).

where A i s a proper subgroup of P(p) . The corresponding

2-closed penrmtation groups E S are described i.n 3.4(2).

Clearly, a l l the S-rings are palrnise non-S-isomorphic.

Moreover, P(p) ' 1s the s e t of all S-isomorphiams 1 nontr ivia l S-r im onto i t s e l f .

(This follows from the f a c t that V: X i+ xa(la)-I (x€Zp) i e

an S-isomorphiam f ixing l€Zp. Yherefore V muet be an

S-automorpbiam f ixing 1 , consequently V = 0 and

15 ~ ( l ~ ) * = ( l ~ ) ~ € l P ( p ) ' . ) . I (cf. [238 8.4.12], [22]).

4. The isomor~hism pmblem

4.1_ The isomorphism problem f o r cyc l i c graphs 1s of grea t

importar.ce f o r appl ica t ions ( f o r concrete r e e u l t s see , f o r

ins tance (35, 36, 371). Aithough we s h a l l not t r e a t quea-

t i o n s concerning p rac t i ca l relevancy, we want t o mention a

l i t t l e example :

Two given ~ y s t e m s of n l i n e a r (o r boolean) equations can be

ca l l ed t o be isomorphic ( t o be "nearly the saueqq) i f the

matrices of coe f f i c i en t s can be obtained fmm each o the r by

pemut ing rows and colums s iml taneously . Then a l s o the so-

l u t i o n s - o r the invers matr ices - of these systems can be

computed e a s i l y from each o the r by corresponding pemuta-

t ions. Thus we can speed up the computation of so lu t ions of

(systems o f ) equations i f we would know how t o decide: Are

two matrices isomorptic? - This pmblem turns out t o be

the isomo~phiem problem f o r cyc l i c graphs i f we consider

t he oft?n used ao-called c i r cu lan t matr ices (cf .[7],[35-371)

which are exact ly the adjacency matrices

E = (aij)i,j=, ,". ,n , (ai j=l 4=+j-iel'),

of cyc l i c graphs I? =Zn. Of Course, the simpler the so lu t ion

of the isomorphimn pmblem the b e t t e r 1s the app l i cab i l i t y .

For cyc l i c graphs, the simplest case 16 given i n the

f o l l o r i n g theorem: . ;

4.2 1 8 o ~ o % g ~ g ~ ~ ~ g g g g ~ - O o % - c ~ ~ = 1 = i ~ = ~ g r c g k ~ ~ 2 ~ ~ ~ Z P'

Let n = p & a prime. Two c y c l i c graphs P, P15Zn a r e isomor- - phic i f f there e x i e t s an m ~ P ( n ) aith T ' = P m. 1([1 ,6,28,7],

nota t ions cf . 1.3. 1.15).

Unfortunately. the m g e c t u - e of A. k ~ n , namely t h a t 4.2

holde f o r all n r m , was not t r u e i n gene&l[7]. Neverthelese,

t he following-theorem shows t h a t t he re can e x i s t very simi-

liar isomorphism conditione:

Let F, P ' = Z (P prime) be c y c l i c graphe and - P

r ( p ) : = i ~ c r J g.~ .d . (~ .p)=p] .~( , ):= r \ F ( ~ ) ,

(analogeouelg for I! ). Then I! g& F E isomoruhic iff 2 t he re e x i s t mo,ml E P ( p ) such t h a t --

I (Ei) = )Do - and P'(p) = P(p)ml where I I (b) mo =ml (fv~dfun-con.jecture -I1) pmvided that

i ( i ) ) I = P ~ - P z I

( i i ) j r ( p ) l = P - 1

( i i i ) P(1 ) ( l+p) # ),(i.e., 3 k e ' ~ ( ~ ): k ( l + ~ ) j P ( ~ ) ) . l ( aee

J [i7],[23; 8.5.14). r.1 I n t h e terminology of ~ . ~ a b a i [3.4,5] we ge t (Cf .1.14.1.15):

t i

Corollary( [6,28.3] 1. Z and Z4 grou~t i . W P -

. Y ~oml ia rp - . z 2(pf2 prime) i s not 9-CI. W(ai. M. too).

P

Together with the rnethod of S-rings, the following theo-

rem i s Fundamental f o r the inveetigation of isomorphisms of

Cayley graphs..In par t icular , the above theorems 4.2, 4.3

and our isomorphim theorem 4.7 f o r cyclic graphs over Z P9

can be proved by t h i s theorem. Moreover, the follorring theo-

rem can give important h in t s f o r possible isomorphim c r i -

t e r i a of Cayley graphs.

436 Theorem. Let (H; ,e) & a w u p auch tha t no two in-

duced (2.4) S-rings - H & S-isomorphic. s (H;&) & = Cayley ßraph of H G:= Aut,ZC ( 2 H'). W: ( i ) For every isomomhism v:(~;id-+(Ei;Z*) onto a Caglex

m Z ' s H & &

VC%(G) & ~ = a a * = : G ' .

( i i ) There ia 1-1-correspondence between the Cayley grauhs

T' H isomomhic & end the elements (= r inh t cosets Gv)

of the fac to r nroup R (G)/G. (Gv ia the s e t of all isomer- --- %i

phisms f m m a onto the a* s s 1.

( i i i ) [I? (G) : G] hi the number of a l l Cavley m p h s 6 ' H %I

abich are isomorphic 5 E.

w o O ( c f . 23 ). ( I ) : Beceuse G ' = ~ - ~ ~ v a n d H ^ G'we get

~ ~ = ( h ' ) - ~ G ' h ~ = W - ~ G W f o r az -vhA, h:=(ev)", and therefore

G;- w-'Gew eince ew=e. We show tha t W i s an S-ieomorphism

between s:=s(G,H) and S':=S(G*,H) (cf. 2.5, 2.2):

W=~-^W= z w g ' I (for Y E T S ~ ) ~ +> y=zßI (g.~,) W Y

g1 :=w-~-Aw((P )-' 1.1 0$ jW € ~ 7 i ~ ) x ~ . (remark tha t

-33-

g P e n - ' ~ x ^ w ( ( x ~ ) - ' ) ^ = ~ ~ , (HAfiG,G1), and -1

eg'=((ew ~ ~ x ~ w ~ x w ~ - l = ~ x w ~ (xW)-'= e , i.e. g * a ~ * ~ ) .

Thus g(G ,H)=g (G ,H) because S-isomorphic (induced) S-ringe

are equal. G and G 1 are 2-cloeed, therefore we have G = G 1

by 2.6(1). Now, veN (G) followe f r o m G=G~=V-'GV. 51

( i i ) : By (I ) , i t d f i c e s to remark that fl= E' f o r two

ieomorphiams v,v9 i f f v1v- 'e~, i.e. vlcGv.

( i i i ) d i rect ly follows f r o m ( i i ) . D

1.n U1 the caeee H = Z „ n&lpq,p2], thq condition of

pairr iee non-5-ieonorphic S-ringe i e f u l l f i l l e d . in general,

it. i e an

p?gp,gm_glgm, detellnine all groupe H such thet a i l S-ringe (or

induced S-rings, reep.) over H 81% painrise non-S-iso-

morphic . Looking at the proof of 4.6 it would d f i c e t o have t h i s

property i o r all S-ringe induced by a König group (QS ge-

nerated by one eimple quantity! - provided H i e a Schur

group, cf. 3.2).

Tfie following theorem provee that Z i e a 9-CI D U D ~ ~ ] P9 - -

what independently was found recent ly by C.D. ~odeil[9], too.

Let n = pq (p L q pri i ies) . c v c l i c maphs F, T ' s Z n - isomorphic i f f t he r e e x i s t s an meYP(n) auch t h a t

F < = Fm ( = { m ( z e T ] ) .

4.8 Example, The cyc l i c graphs T ={ 1,2 ,9) and F' ={3,4,7)

over Z1 given i n 1.1 6 a r e iso!norphic because F' = F 7

(7eIP(10)) . The mappim V: X H 7x (mod 10 ) is an ieomor-

phiEU~.

We a r e going t o prove t h e isomorphiam theoren.

pwwg 2: &J. += I': If T ' = Tm then t he mapplng

V: X c+ xm (mod n ) i a nn isomorphiam (X-ytF ++ xrn-ymcF ' 1.

lB+": Let C:= = ' F (G i e a König group!) and l e t

V: Zn--+ "„ be an isomorphiam from i? onto a c y c l i c graph F'.

We know vtN ( G ) and u s ine 4.6(11) i t remains t o prove t h a t 'n

f o r a l l poss ib le (i.e. König) p o u p s C t he r e e x i s t s an

ma tCv with m€P(n) . For t h i s reaeon we w i l l look a t t h e

König groups G given i n 3 . 3 ( i i i ) , f i n d f o r each C the form

o f viNS (C) and detennine mrlP(n): n

(&gpg-L331: C&,. yhen r=r l= Z n , m:=leP(n) .

C;g~~g12,~-b~: Case 3.4(1 bl ) : C = (;'=C''. Then

(GS)"liS (G") by 1 .Y. l e d i s t i ngu i sh 3 subcnses: n P

5 ) C1=S arid C1'=Sp 9 (cf. 3 . 4 ( 2 ~ ) ) . Then [Iis n (c ) :G]=~

and we obta in T ' = F by 4 .6 ( i i i ) ; m:=1 e P ( n ) .

d2) G'=S and G"=AMZA 9 P

(Alv (& Zp). Then vrNS (G) = n

= SqxlP(p)"ZA has the form (g' ,sAb"), g16Sq, sf lP(p) , P

btZp. Chose dtlF'(n)Asuch that m = e mod p (m:=se +e 1. 9 P

Then mArGv. The case G1*=S i s to t r e a t anaiogeously. P

p) G ' = A * Z ~ arid G " = A N Z ~ (AV" z zp.(3.rtzfir 9 P 9'

Then ?i (G)= HI(Zq)*H(Zp)= H(Zn) by 1.8 and we can

proceed a s i n the following

Case 3.4(lb2): F = A ^ Z L (A<IF(nj!. Then N (G)=H(Zn) Sn

by 1.4 and every vf NS (G) i s li:i&r ( c r : 3 ) : v=m^bA , n

m" ~ p ( n ) ^ , bAeZi. Since Zif G we have 9-=~9"-le VG=GV

where mclP(n).

Groupe 3.3 cl ): G = G'2Gw (G159„ ""'" .I 'I *"o oBnig groups 1. --------------

By 1.6, we have (notation eee i . 5 )

If both G ' and G'' are a s descrjhoc! ru ?-4(28) then

" - HI(Z ) , therefcye vcH (G) is Ns (G')=H(Zq) and N (G )- 9 S~ P

of the form S a * ~

V: (x,y) H (xu,$) d t h U = aAbA and rx = c;dxA,

( a a e P ( q ) ^ , bAeZ;, c X A ~ P ( p ) " , dxAe~;l (x,y)ezq*Zp).

Since ZA t G ' , Z" s G W attd wx-' wo E Gw f o r a l l xcZ (O=rero~Z ), 9 P 9 9

the permutation -1

. 7: (x,y) » (f (X) , (Y + axl belonge t o G = G ' Z G 1 t f o r a l l fdZ;, S6zp . For f:=(-i-lb)",

-1 i.e. f (x> := X - a-'b. and P:= df - df(x)cO we get e a s i l r

? V : (x,y) C, (axlc0y), i.e.* h = m A where m-ae +C e P 0 9'

and we have mAeCv since %C; (miP(n) ).

For the other cases, namely i f GS=S o r Cw=Sp, one can f i n d 9

eas i ly a f O r C such t h a t V'=V~VSGV has the same form a s the

above V; thus an mA6B(n)^ with m A = ~ l e ~ v S = ~ v a l so exia ts .

Groups 3.3 cp): C=GtSZG'. lohe proof of this case rune -10- -------------- geously t o the preceding one by changing p and q.

The proof i s finiehed non, because a l l König p o u p s have

been considered. I

A group H =$I; ,e) may be not a $-CI p o u p ( fo r instance

Z z i 4.5), nevertheleas there can exia t a - l o t of CI-graphs P

of H ( fo r instance 4.3(b)) f o r which we have good isomor-

phism c r i t e r i a (cf. 1.1 4). Therefore, a t the end of t b i s

aection, we want t o look f o r CI-graphs of a group H ( i n

general not CI). For a general chanicterization of Cayley

objecta with the. CI-pmperty aee a lso [3; Lemma 3.11.

Theorem. Let H & g group such tha t no two induced

S - r i n ~ a over H S-isomorphic. Then a Cayley mach 6 H

i s 2 CI-maph iff - [ N ~ ~ ( G ) : G] = [ N ~ (H- : N ~ ( H - I]

H where C = A u t , B . (Remark: For the above equation it auff ices

*

t o ahow tha t the l e f t aide i a not greater than the r i g h t ond.

Eote. t h a t [E H : GhA& H] i a the number of a l 1

cayiey graphs 6 ' isomorphic t o 6 with reapect t o an i aomor

phism VSG H. We have:

[E 11 : H]= ((1- ti)iin: (in*- H)H-]=[A. ( I ~ ) : E ~ ( H ~ ) ] 5h

( b e c a ü ~ e of 1.2 end i1-sC). Together wi th 4 . 6 ( i i i ) , t h e

theorem i a yroved. I

Yl'lieorem 4.3 showa - among o t h e r t h ings - t h a t t h e proper- che

t y t o be a CI-graph depends on ly o>automorphiwi gmup o f

t h e graph (c f . also[3;~emma 3.11 ). We ge t :

4.10 Coro l l r ry . Let 11 be a s i n 4.9 and l e t u s c a l l a

ktinic group (G,li) 2 (iia,n) t o be an (i.e. t h e

"Ad.dam-conjecture" is f u l l f i l l e d f o r graphs 6 wi th &J iL =G) ,

i f some Cayley grnph B of H (and t he r e fo r e a l l ) wi th

G i s a CI-paph o f H. Then t h e fol lowing holds:

I f - i iA$ G f H(H)

then ü i s an Adam nroup (& p a r t i c u l a r , Hn i t s e l f i e then - --- a n j\ti?m -0. --

iymof. Yor iiAgG $ I i ( I l ) = ( & H)Ha we have (as a ~ l i g h t gene-

r s l i z a t i o n oi 1.4): li ( C ) $ P . (tia) (because HL i s t h e on ly %I SH

r e m l a r subgroup of Ii(h) o f degree ( I i l ) . Therefore

LiTF ( ~ ) : G ] ~ [ i i ~ ~ ( ~ " ) : G n l t ( ~ ^ ) l = I h (I~"):II~(I! ' ) ] , and G is n II

a n IdFm group by 4.9. 1 SI1 .

I n connection with - C I groups t h e r e a r i s e s t h e fo l lowing 9

g~g~,ga961=c@ :

Under which condi t ions the d i r e c t product H'*H1' of

9 - C I groups H ' and H" 1s again 2 1 - C I group? 1 (e.g., Zp, Zp*Zp 137, ZpxZq and Z x ... 8 Z with some addi-

P1 P r t l o n a l condit ions f o r tke primes p, , ... ,pr L201 are C I groups , Cf. [5] 1.

4.11 Remarks.

a ) From theorem 4.9 one con derive some condit ions under

which the d i r e c t product of CI-graphs (o r k d h groups) i s

again a CI-graph (o r Id6m group), but t he r e s u l t s a r e s t i l l

f a r f r o m the so lu t ion of the above Open problem.

b) A group H i e ca l l ed a C ~ - p ~ 1 [ 3 , 2 0 ] i f between any two

ieomorphic r e l a t i o n a l s t ruc tu re s on the group G ( a s under-

l y lng s e t ) t he automorphism groupeof which contain H" t he re

e x i s t s an isomorphiem which i s a t the saue time an automor-

p h i m of H (cf. 1.14). 1n[20] it 1s shown t h a t - besidea the

group H of o d e r 4 - t he f i n i t e C I p o u p s must be c y c l i c

of order n euch t h a t y(n) ( ' pEu le r ' s funct ion) is Prime

t o n. Some e u f f i c i e n t (possibly a l so necessary) condi t ions

f o r a group t o be a C I group a r e given.

Considering the above property f o r H only f o r r e l a t i o n a l

s t ruc tu re sn i th a t most r-ary r e l a t i o n s ( r t N ) we ge t the

not ion of a ~CE-grogE H.

such r e s t r i c t i o n s seem t o make t h e s i t u a t i o n much more

d i f f i c u l t . For ins tance ,

z ~ q is 2-CI (4.7, 3.31, but not CI if g.c.d.((p-1 )(q-1 ),pq)/

f i ,[20]i

Zpp is 3-CI but not 4-CI, [3] . Clearly, r-CI implies r'-CI for r'jr.

We state here the following

g g e , g , ~ E a g :

Characterize the finite r-CI groups, (r22) !

The niethod of Schur, namely the investigation of S-rings,

is one of the tools to solve this problem for r = 2.

lany important result s concerning the description

of %-CI groups (using group theoretic methods) are to

be found in /3,4,5].

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