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A Classification of DCI(CI)-Subsets for Cyclic Group of Odd Prime Power Order
Journal of Combinatorial Theory, Series B 78, 2434 (2000) doi:10.1006jctb.1999.1924, available online at http:www.idealibrary.com on
A Classification of DCI(CI )-Subsets for Cyclic Group of Odd Prime Power Order 1 Qiongxiang Huang and Jixiang Meng Department of Mathematics and Institute of Mathematics and Physics, Xinjiang University, Urumuqi, Xinjiang 830046, People's Republic of China E-mail: huangqxxju.edu.cn Received July 15, 1998
Let Z n be a cyclic group of order n and C(Z n , S) the circulant digraph of Z n with respect to S Z n "[o]. S is called a DCI-subset of Z n if, for any circulant digraph C(Z n , T ), C(Z n , S)$C(Z n , T ) implies that S and T are conjugate in Aut(Z n ), the automorphism group of Z n . In this paper, we give a complete classification for DCI-subsets of Z n in the case of n= p a, where p is an odd prime. 2000 Academic Press
Key Words: circulant digraph; isomorphism; DCI-subset.
1. INTRODUCTION For a finite group G and a subset S G "[1], a Cayley digraph of G with respect to S is defined by C(G, S)=(V, E), where V=G and (u, v) is an arc in E if and only if vu &1 # S. If S=S &1 ( =[s &1 | s # S]), C(G, S) may be viewed as a Cayley graph. If G=Z n , C(Z n , S) is generally said to be a circulant digraph. Let A=Aut(C(G, S)) be the automorphism group of C(G, S). Then R(G)=[R(g): a Ä ag(\a # G) | g # G] is a subgroup of A, which acts transitively on the vertices of C(G, S). Let Aut(G) be the automorphism group of G. A subset S G "[1] is called a DCI-subset of G if, for any C(G, T ) isomorphic to C(G, S), there exists a : # Aut(G) such that T=S : (i.e., T is conjugated to S in Aut(G)). Otherwise S is a NDCI-subset. A DCI-subset S will be referred to a CI-subset if S=S &1. Evidently, C(G, S)$C(G, T ) if S and T are conjugate in Aut(G). in 1967, Adam in [1] proposed a conjecture that C(Z n , S)$C(Z n , T ) if and only if S and T are conjugate in Aut(Z n ), which initiated the investigation of isomorphism for circulant digraphs, and further the Cayley digraphs. Regretfully, however, several counterexamples in [2, 3] disproved Adam's conjecture. In fact, there 1 Supported by the National Nature Science Foundation of China and Morningside Center of Math., Chinese Academy of Science.
24 0095-895600 35.00 Copyright 2000 by Academic Press All rights of reproduction in any form reserved.
CLASSIFICATION OF (DCI )(CI )-SUBSETS
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exists a large number of pairs of subsets S and T in some G which are not conjugate but C(G, S)$C(G, T ). Up to now the necessary and sufficient condition for the isomorphism of Cayley digraphs has been concealed. It seems difficult and has a long way to fully characterize the isomorphic classes of Cayley digraphs. But there are still some profound results obtained particularly in the characterization of DCI-subsets, which is now the main subject in the investigation of this isomorphic problem. One can see [118] for references. A finite group G is called a DCI-group if any subset S G "[1] is a DCIsubset. Otherwise G is a NDCI-group. It is believed that the bulk of groups are NDCI-groups. But, as we know, there is only one NDCI-group Z p 2 whose DCI-subsets are exhausted in [7] by B. Alspach and T. D. Parsons. In this work, we devote ourselves to the classification of DCI-subsets of Z p a , where p is an odd prime. For convenience, the cyclic group Z n is regarded as the additive group of integers modulo n. Its operation is denoted by +, unit by 0 and &u the reverse of u in Z n . d # Z n generates a subgroup ( d) and ( d) +u is a coset of ( d) containing u # Z n . Let S Z n "[0] and S generates Z n . If there exists m | n (1
S 1 = s1 # S1 ((m) +s 1 ) (that is, S 1 is the union of some cosets of S 2 /( m).
Then S=S 1 _ S 2 is said to be a m-partition of S. Let us consider Z p a . First we introduce some notations and definitions which will be used in the following sections. Let S Z p a "[0]. s # S is called a coset element of S if ( p a&1 ) +s S. Otherwise s is a non-coset element of S. Now we divide the generating subsets of Z p a into three parts: A generating subset S Z p a "[0] is said to be I-type if S contains a non-coset element s with p |% s, III-type if S has a p r-partition (0
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HUANG AND MENG
2. PRELIMINARIES In this section, we shall quote and give some preliminary results for the purpose of characterizing the DCI-subsets of Z p a . The first result due to Babai [5] is a criterion for a subset of a finite group to be a DCI-subset. Lemma 2.1 (Babai [5]). Let C(G, S) be a Cayley graph of finite group G with respect to S G"[1], and A=Aut(C(G, S)). Then S is a CI-subset of G, if and only if, for any { # Sym(G) (the symmetric group on G) with R(G) { A, there exists an : # A such that R(G) { =R(G) :. Let A=Aut(C(G, S)), N A (R(G))=[: # A | R(G) : =R(G)] the ``normolizer'' of R(G) in A and Aut(G, S)=[: # Aut(G) | S : =S]. The next result due to C. D. Godsil [6] is helpful in the use of Babai's criterion. Lemma 2.2 (Godsil [6]). Let C(G, S) be a Cayley graph of finite group G with respect to S G "[1] and A=Aut(C(G, S)). Then N A (R(G))= R(G) < Aut(G, S), where < denotes the semi-direct product. It is generally known that the above two results are also valid for Cayley digraphs, which means that Babai's criterion is suitable to the judgment of DCI-subset of G. The following result is a consequence of Proposition 4.1 in [16], the conclusion of which is drawn for CI-subsets. Here, for the integrity, we prefer to give its proof for DCI-subsets. Lemma 2.3. Let C(G, S) be a Cayley digraph of finite p-group G with respect to S G "[1]. If p |% |Aut(G, S)|, then S is a DCI-subset of G. Proof. Let A=Aut(C(G, S)). If R(G) is a Sylow p-subgroup of A, then it immediately gets our result by Sylow Theorem and Lemma 2.1. Otherwise suppose P is the Sylow p-subgroup of A with P>R(G). Then by Lemma 2.2 R(G)
CLASSIFICATION OF (DCI )(CI )-SUBSETS
27
Proof. Let G*=.R(G) . &1. Then G* A=Aut(C(G, S)). Let (u, v) be any arc of C(G, S 1 ), that is, vu &1 # S 1 . It is easy to check that :=R(u) &1 .R((u . ) &1 ) . &1 # A 1 . We have (vu &1 ) : # S 1 , i.e., ((vu &1u) . (u . ) &1 ) . # S 1 , that is v . (u . ) &1 # S .1 =T 1 . Hence . is an isomorphism from C(G, S 1 ) to C(G, T 1 ). Similarly one can prove that . is an isomorphism from C(G, S 2 ) to C(G, T 2 ).
3. ISOMORPHISM OF CIRCULANT DIGRAPH OF LEXICOGRAPHIC PRODUCT Let 1 1 =(V 1 , E 1 ) and 1 2 =(V 2 , E 2 ) be two digraphs(or graphs). The lexicographic product of 1 1 by 1 2 is a digraph(or graph) 1=1 1[1 2 ] with vertex set V 1 _V 2 and ((u 1 , v 1 ), (u 2 , v 2 )) is an are (or an edge) if and only if either (u 1 , u 2 ) # E 1 or u 1 =u 2 and (v 1 , v 2 ) # E 2 . Let S Z n "[0]. Suppose S has a m-partition S=S 1 _ S 2 . Then by definition S 1 = . (( m) +s 1 )
and
S 2 ( m).
s1 # S1
It is easy to check that C(Z n , S) is the lexicographic product of C(Z m , S 1 ) by C(Z nm , S 2 ), where S 1 =S 1 ( m) Z n (m) =Z m and S 2 =[s 2 m | s 2 # S 2 ] Z nm . Recall that S generates Z n therefore S 1 {<. If S 2 =<, then C(G, S)=C(Z m , S 1 )[K cnm ], where K cnm is the complement of the complete graph K nm , which will be called the coset circulant digraph. A m-partition S=S 1 _ S 2 is said to be maximal if there is no m$ | m (1
(1)
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HUANG AND MENG
In fact, if |S 1 | =1 then |N + 1 (x) & S| = |S 2 | < |S 1 | for any x # S 1 . We now suppose |S 1 | 2 in what follows. It is easy to see that there exists v # S 1 such that v adjoins to at most |S 1 | &2 vertices of S 1 in 1 1 , since otherwise 1 1 will be a complete graph. Hence |N + 1 | |S 1 | &2. This implies 11(v ) & S that for any x # (m) +v S 1 (where v =( m) +v) |N + 1 (x) & S|
n n . ( |S 1 | &2)+2 |S 2 | = |S 1 | +2 |S 2 | & m m
\
+
Note that S 2 /( m), we have |N + 1 (x) & S| < |S 1 |. On other aspect, for any y # S 2 /( m) |N + 1 ( y) & S| |S 1 |.
(2)
Let 2 1 , 2 2 , ..., 2 d be some orbits of A 0 acting on S, each of which contains at least an element of S 2 and S 2 1 i d 2 i /S. Since C(Z n , S) is the lexicographic product of C(Z m , S 1 ) by C(Z nm , S 2 ), the action of A 0 must be transitive on ( m) +s 1 for s 1 # S 1 . This means that ( m) +s 1 2 i if 2 i & (( m) +s 1 ){< (1 i d ). "( 1 i d 2 i ). Then S * Let S *=S 1 1 is the union of some cosets of ( m), and by (1) and (2), S * {<. Additionally, we have S*1 { =S * and 1 1 { ( 1 i d 2 i ) = 1 i d 2 i for { # A 0 . Now we consider 1 *=C(Z which is clearly a coset circulant 1 n , S *), 1 digraph. Set K*=[u # Z n | u+S 1*=S 1*]. It is simple to show that K* is a subgroup of Z n and ( m) K*. Notice that the vertices of K* are just all the vertices which are out adjacent in C(Z n , S *). We have K* { =K* for to every vertex of S * 1 1 : # Aut(C(Z n , S *)) | 0 =0]. By Lemma 2.4, A 0 A 0* , and hence { # A *=[: 0 1 K* : =K* for : # A 0 . Since 2 i is an orbit of A 0 and ( m) K*, we deduce is a m$-partithat 2 i K* (i=1, 2, ..., d ). It implies that S=S * 1 _ (S "S *) 1 tion, where K*=( m$). Since S=S 1 _ S 2 is maximal, we claim that K*=( m). Hence, for any { # A 0 , S {2 =S 2 and so S {1 =S 1 . We thus complete this proof. Lemma 3.2. Let S Z n "[0]. If S=S 1 _ S 2 is maximal m-partition, then C(Z n , S)$C(Z n , T ) if and only if there is maximal m-partition T=T 1 _ T 2 such that C(Z n , S 1 )$C(Z n , T 1 ) and C(Z n , S 2 )$C(Z n , T 2 ). Proof. Since C(Z n , S) is the lexicographic product of C(Z m , S 1 ) by C(Z nm , S 2 ), where S 1 =S 1 (m) Z n ( m) =Z m and S 2 =[s 2 m | s 2 # S 2 ] Z nm , the sufficiency is evident. Now we prove the necessity.
CLASSIFICATION OF (DCI )(CI )-SUBSETS
29
If C(Z n , S) is a complete graph, then C(Z n , T ) is also a complete graph, and hence our result is valid. If S 2 /( m) and S 1 =Z n "( m), then C(Z m , S 1 ) is a complete graph. Let S c =Z n "(S _ [0]) and T c =Z n "(T _ [0]). Then C(Z n , S c ) and C(Z n , T c ) are the complements of C(Z n , S) and C(Z n , T ) respectively. Since C(Z n , S)$C(Z n , T ), C(Z n , S c )$C(Z n , T c ). Clearly C(Z n , S c ) is several copy of C(( S c ), S c ). Hence C(( S c ), S c )$C(( T c ), T c ).
(3)
Notice that Z n is a cyclic group, ( T c ) =( S c ) ( m). This implies that T has maximal m-partition: T=T 1 _ T 2 with T 1 =Z n "( m) and T 2 ( m). From (3), C(Z n , S 2 )$C(Z n , T 2 ). In addition, both C(Z m , S 1 ) and C(Z m , T 1 ) (T 1 =T 1 ( m) ) are complete graphs, C(Z n , S 1 )$C(Z n , T 1 ). At last we suppose S 1 {Z n "( m). Let . be an isomorphism from C(Z n , S) to C(Z n , T ) with 0 . =0. Set T 1 =S .1 and T 2 =S .2 . Then by Lemma 3.1 and Lemma 2.4, C(Z n , S 1 )$C(Z n , T 1 )
and
C(Z n , S 2 )$C(Z n , T 2 ).
For any isomorphism f from C(Z n , S) to C(Z n , T ) with 0 f =0, &1 =S 1 .f &1 # A 0 =[: # Aut(C(Z n , S)) | 0 : =0]. Again by Lemma 3.1, S .f 1 and S 1f =S .1 =T 1 . Let u # (m) and f =R(u) .R(&u . ). Then f is an isomorphism from C(Z n , S) to C(Z n , T ) with 0 f =0. Thus we have .
.R(&u ) =S 1f =T 1 O (S 1 +u) . =T 1 +u . O S .1 =T 1 +u . O S R(u) 1
T 1 =T 1 +u .
for any
u # ( m).
(4)
Let K=[v # Z n | v+T 1 =T 1 ]. Then K is a subgroup of Z n and T 1 = t1 # T1 (K+t 1 ). By (4), ( m) . K. Note that each vertex of ( m) is out-adjacent to every vertex of S 1 in C(Z n , S 1 ) and each vertex of K is outadjacent to every vertex of T 1 in C(Z n , T 1 ). Additionally, m-partition of S=S 1 _ S 2 is maximal, there is no other vertex which is out-adjacent to every vertex of S 1 in C(Z n , S 1 ) except that in ( m). We deduce ( m) . =K, i.e., (m) . =( m). It then follows T 1 = t1 # T1 (( m) +t 1 ) and T 2 =S .2 ( m). Hence T=T 1 _ T 2 is a m-partition which is clearly maximal. We complete the proof. It is well known that Aut(Z n )=[* # Z n | gcd(*, n)=1] and * # Aut(Z n ) is defined by *: u Ä *u(mod n),
for
\u # Z n .
Hence Aut(Z n , S)=[* # Aut(Z n ) | *S=S] Aut(Z n ).
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Theorem 3.1. Let S Z n "[0]. If S=S 1 _ S 2 is maximal m-partition, then S is a DCI-subset of Z n if and only if both S 1 and S 2 are DCI-subsets of Z n and (Aut(Z n , S 1 ), Aut(Z n , S 2 )) =Aut(Z n ). Proof. If S=Z n "[0], C(Z n , S) is a complete graph, and our result clearly holds. In the following we assume S/Z n "[0]. We first prove the sufficiency. Let C(Z n , S)$C(Z n , T ) and . the isomorphism from C(Z n , S) to C(Z n , T ) with 0 . =0. Set T 1 =S .1 and T 2 =S .2 . By Lemma 3.2, T=T 1 _ T 2 is maximal m-partition, C(Z n , S 1 )$ C(Z n , T 1 ) and C(Z n , S 2 )$C(Z n , T 2 ). According to assumption that S 1 and S 2 are DCI-subsets of Z n , there are * 1 and * 2 in Aut(Z n ) such that T 1 =* 1 S 1 and T 2 =* 2 S 2 . Since ( Aut(Z n , S 1 ), Aut(Z n , S 2 )) =Aut(Z n ), there exists ' i # Aut(Z n , S 1 ) (i=1, 2, ..., d) such that Aut(Z n )=' 1 Aut(Z n , S 2 ) _ ' 2 Aut(Z n , S 2 ) _ } } } _ ' d Aut(Z n , S 2 ). Thus there is ' i (1 i d ) such that ' i Aut(Z n , S 2 )=* &1 1 * 2 Aut(Z n , S 2 ) O Aut(Z n , S 1 ) & * &1 1 * 2 Aut(Z n , S 2 ){<. Hence there is * # Aut(Z n ) such that * # * 1 Aut(Z n , S 1 ) & * 2 Aut(Z n , S 2 ). It follows that *S 1 =* 1 S 1 =T 1 and *S 2 =* 2 S 2 =T 2 , that is, T=*S and so S is a DCI-subset of Z n . Conversely, let S/Z n "[0] be a DCI-subset of Z n and S=S 1 _ S 2 maximal m-partition. Suppose C(Z n , S 1 )$C(Z n , T 1 )
and
C(Z n , S 2 )$C(Z n , T 2 ).
Set T=T 1 _ T 2 , which is clearly maximal m-partition. Then by Lemma 3.2, C(Z n , S)$C(Z n , T ), and hence there is * # Aut(Z n ) such that T=*S. It follows that T 1 =*S 1 and T 2 =*S 2 . Therefore S 1 and S 2 are DCI-subsets of Z n . In addition, taking T=T 1 _ T 2 , where T 1 =* 1 S 1 and T 2 =* 2 S 2 (* 1 , * 2 # Aut(Z n )), similarly we get C(Z n , S)$C(Z n , T ), and hence there is * # Aut(Z n ) such that T=*S, i.e., *S 1 =* 1 S 1 and *S 2 =* 2 S 2 . So we have * # * 1 Aut(Z n , S 1 ) & * 2 Aut(Z n , S 2 ), that is, Aut(Z n , S 1 ) & * &1 1 * 2 Aut(Z n , S 2 ){<.
(5)
Let + # Aut(Z n ) be arbitrary. By (5), Aut(Z n , S 1 ) & + Aut(Z n , S 2 ){<. Hence there exist + 1 # Aut(Z n , S 1 ) and + 2 # Aut(Z n , S 2 ) such that + 1 =++ 2 , so +=+ 1 + &1 2 . It completes the proof.
CLASSIFICATION OF (DCI )(CI )-SUBSETS
31
4. CLASSIFICATION OF DCI-SUBSETS OF Z p a Let p be an odd prime. In this section we shall completely characterize the DCI-subsets of Z p a . It is well known that Aut(Z n ) is an abelian group of order .(n), where . is the Euler function. In particular, for n= p a we have the following. Lemma 4.1. Let p be an odd prime. Then Aut(Z p a ) is a cyclic group of order p a&1 ( p&1) and ( p i ) +1 is the unique subgroup of order p a&i (0
(6)
Then 1 k
and
S 1 =S "S 2 .
(7)
By definition each s 1 # S 1 is a coset element. We call j the coset index of s 1 in S 1 if ( p j ) +s 1 S 1 and ( p j&1 ) +s 1 3 S 1 . It is clear 1 j a&1. Set t=max[t | j is the coset index of s 1 and s 1 # S 1 ].
(8)
By definition ( p t ) +s 1 S 1 for s 1 # S 1 and hence S 1 = s1 # S1 (( p t ) +s 1 ). Additionally k
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HUANG AND MENG
Lemma 4.2. Let S be a II-type subset of Z p a (a 3) and S=S 1 _ S 2 is defined in (7). Set T=T 1 _ T 2 , where T 2 =S 2 and T 1 =*S 1 for * # ( p t&k ) +1 (k and t are defined in (6) and (8)). Then C(Z p a , S)$ C(Z p a , T ). Proof. Let u=x 1 p k + y 1 # Z p a * # ( p t&k ) +1, let us define f * as
(0 x 1 < p a&k, 0 y 1 < p k ).
For
f * : u Ä u f* =x 1 p k +*y 1 . It is routine to verify that f * is an one to one mapping on Z p a . Let v=x 2 p k + y 2 # Z p a (0 x 2 < p a&k, 0 y 2 < p k ). Then v f* &u f* =(x 2 &x 1 ) p k +*( y 2 & y 1 )=(x 2 &x 1 ) p k (1&*)+*(v&u).
(9)
Suppose (u, v) is an arc of C(Z p a , S). It suffices to show that (u f*, v f* ) is an arc of C(Z p a , T ). If v&u=(x 2 &x 1 ) p k +( y 2 & y 1 ) # S 2 , y 1 = y 2 and hence v f* &u f* = (x 2 &x 1 ) p k =v&u # S 2 =T 2 . Suppose v&u # S 1 . Let j be the coset index of v&u. Then ( p j ) +(v&u) S 1 O ( p j ) +*(v&u) *S 1 =T 1 .
(10)
By definition of t in (8), t j and ( p t ) ( p j ). Since * # ( p t&k ) +1, 1&* # ( p t&k ) and hence (x 2 &x 1 ) p k (1&*) # ( p t ) ( p j ). By (9) and (10), we have v f* &u f* # ( p j ) +*(v&u) T 1 . It completes the proof. Theorem 4.2. Let S be a II-type subset of Z p a ( 3). Then S is a NDCI-subset of Z p a (a 3) (i.e., S is not a DCI-subset). Proof. Let S=S 1 _ S 2 be a II-type subset of Z p a , which is defined in (7). Let T=T 1 _ T 2 , where T 1 =*S 1 (* # ( p t&k ) +1) and T 2 =S 2 . By Lemma 4.2 C(Z p a , S)$C(Z p a , T ). To the contrary suppose S is a DCI-subset. Then there is ' # Aut(Z p a ) such that T='S, that is, 'S 1 =*S 1 and 'S 2 =S 2 O ' # * Aut(Z p a , S 1 ) & Aut(Z p a , S 2 ). It implies for any * # ( p t&k ) +1 * Aut(Z p a , S 1 ) & Aut(Z p a , S 2 ){<.
(11)
Remember that S 2 contains a non-coset element s* with p k & s*. We claim that ( p a&k&1 ) +1 Aut(Z p a , S 2 ).
(12)
Since otherwise (( p a&k&1 ) +1) s*=( p a&1 ) +s* S 2 , which contradicts the assumption of s*.
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CLASSIFICATION OF (DCI )(CI )-SUBSETS
Since k
(13)
According to the definition of t in (8), there exists s 1* # S 1 such that ( p t ) +s 1* S 1 ,
but
( p t&1 ) +s 1* 3 S 1 .
(14)
Let p l & |Aut(Z p a , S 1 )|. Then ( p a&l ) +1 Aut(Z p a , S 1 ). Suppose p j & s 1* . It is clear 0 j
(15)
Let H$=( Aut(Z p a , S 1 ), Aut(Z p a , S 2 )) and H=( Aut(Z p a , S 1 ), Aut(Z p a , S2 ), ( p t&k ) +1). Then H$, H Aut(Z p a ) and by Lemma 4.1, (13), and (15) we claim that ( p t&k ) +1 H$. Hence H$
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HUANG AND MENG
ACKNOWLEDGMENTS The author thanks the referees for their insightful comments and helpful suggestions, which have simplified the proof of Lemma 2.4 and also improved the presentation of this paper.
REFERENCES 1. A. Adam, Research problem 210, J. Combin. Theory 2 (1967), 393. 2. D. Z. Djokovic, Isomorphism problem for a special class of graphs, Acta Math. Acad. Sci. Hungar. 21 (1970), 267270. 3. B. Elspas and J. Turner, Graphs with circulant adjacency matrices, J. Combin. Theory 9 (1970), 297307. 4. M. H. Klin and R. Poschel, The isomorphism problem for circulant digraphs with p n vertices, preprint, P3480, AdWDDR,ZIMM.Berlin, 1980. 5. L. Babai, Isomorphism problem for a class of point-symmetric structures, Acta Math. Acad. Sci. Hungar. 29 (1977), 329336. 6. C. D. Godsil, On the full automorphism groups of a graph, Combinatorica (1981), 243256. 7. B. Alspach and T. D. Parsons, Isomorphism of circulant graphs and digraphs, Discrete Math. 25 (1979), 97108. 8. C. D. Godsil, On Cayley graph isomorphism, Ars Combin. 15 (1983), 231246. 9. M. Muzychuk, Adam's conjecture is true in the square-free case, J. Combin. Theory Ser. A 72 (1995), 118134. 10. J. Turner, Point symmetric groups with a prime number of points, J. Combin. Theory 3 (1967), 136145. 11. S. Toida, A note on Adam's conjecture, J. Combin. Theory Ser. B 23 (1977), 239246. 12. P. P. Palfy, Isomorphism problem for relational structures with a cyclic automorphism, European J. Combin. 8 (1987), 3543. 13. Q. X. Huang and J. X. Meng, On the isomorphisms and automorphism groups of circulants, Graphs Combin. 12 (1996), 179187. 14. C. H. Li, On the isomorphism of connected Cayley graphs, Discrete Math. 78 (1997), 109122. 15. J. X. Meng and Q. X. Huang, The automorphism groups of minimal infinite circulant digraphs, European J. Combin. 18 (1997), 425429. 16. Q. X. Huang, Isomorphisms and automorphism groups of a class of Cayley digraphs on abelian groups, Australas. J. Combin. 15 (1997), 310. 17. M.-Y. Xu, Automorphism groups and isomorphisms of Cayley digraphs, Discrete Math. 182 (1998), 309319. 18. B. Alspach, Isomorphism and Cayley graphs on abelian groups, in ``Graph Symmetry'' (G. Hahn and G. Sabidussi, Eds.), pp. 122, Kluwer Academic, Dordrecht, 1997.
A Classification of DCI(CI )-Subsets for Cyclic Group of Odd Prime Power Order 1 Qiongxiang Huang and Jixiang Meng Department of Mathematics and Institute of Mathematics and Physics, Xinjiang University, Urumuqi, Xinjiang 830046, People's Republic of China E-mail: huangqxxju.edu.cn Received July 15, 1998
Let Z n be a cyclic group of order n and C(Z n , S) the circulant digraph of Z n with respect to S Z n "[o]. S is called a DCI-subset of Z n if, for any circulant digraph C(Z n , T ), C(Z n , S)$C(Z n , T ) implies that S and T are conjugate in Aut(Z n ), the automorphism group of Z n . In this paper, we give a complete classification for DCI-subsets of Z n in the case of n= p a, where p is an odd prime. 2000 Academic Press
Key Words: circulant digraph; isomorphism; DCI-subset.
1. INTRODUCTION For a finite group G and a subset S G "[1], a Cayley digraph of G with respect to S is defined by C(G, S)=(V, E), where V=G and (u, v) is an arc in E if and only if vu &1 # S. If S=S &1 ( =[s &1 | s # S]), C(G, S) may be viewed as a Cayley graph. If G=Z n , C(Z n , S) is generally said to be a circulant digraph. Let A=Aut(C(G, S)) be the automorphism group of C(G, S). Then R(G)=[R(g): a Ä ag(\a # G) | g # G] is a subgroup of A, which acts transitively on the vertices of C(G, S). Let Aut(G) be the automorphism group of G. A subset S G "[1] is called a DCI-subset of G if, for any C(G, T ) isomorphic to C(G, S), there exists a : # Aut(G) such that T=S : (i.e., T is conjugated to S in Aut(G)). Otherwise S is a NDCI-subset. A DCI-subset S will be referred to a CI-subset if S=S &1. Evidently, C(G, S)$C(G, T ) if S and T are conjugate in Aut(G). in 1967, Adam in [1] proposed a conjecture that C(Z n , S)$C(Z n , T ) if and only if S and T are conjugate in Aut(Z n ), which initiated the investigation of isomorphism for circulant digraphs, and further the Cayley digraphs. Regretfully, however, several counterexamples in [2, 3] disproved Adam's conjecture. In fact, there 1 Supported by the National Nature Science Foundation of China and Morningside Center of Math., Chinese Academy of Science.
24 0095-895600 35.00 Copyright 2000 by Academic Press All rights of reproduction in any form reserved.
CLASSIFICATION OF (DCI )(CI )-SUBSETS
25
exists a large number of pairs of subsets S and T in some G which are not conjugate but C(G, S)$C(G, T ). Up to now the necessary and sufficient condition for the isomorphism of Cayley digraphs has been concealed. It seems difficult and has a long way to fully characterize the isomorphic classes of Cayley digraphs. But there are still some profound results obtained particularly in the characterization of DCI-subsets, which is now the main subject in the investigation of this isomorphic problem. One can see [118] for references. A finite group G is called a DCI-group if any subset S G "[1] is a DCIsubset. Otherwise G is a NDCI-group. It is believed that the bulk of groups are NDCI-groups. But, as we know, there is only one NDCI-group Z p 2 whose DCI-subsets are exhausted in [7] by B. Alspach and T. D. Parsons. In this work, we devote ourselves to the classification of DCI-subsets of Z p a , where p is an odd prime. For convenience, the cyclic group Z n is regarded as the additive group of integers modulo n. Its operation is denoted by +, unit by 0 and &u the reverse of u in Z n . d # Z n generates a subgroup ( d) and ( d) +u is a coset of ( d) containing u # Z n . Let S Z n "[0] and S generates Z n . If there exists m | n (1
S 1 = s1 # S1 ((m) +s 1 ) (that is, S 1 is the union of some cosets of S 2 /( m).
Then S=S 1 _ S 2 is said to be a m-partition of S. Let us consider Z p a . First we introduce some notations and definitions which will be used in the following sections. Let S Z p a "[0]. s # S is called a coset element of S if ( p a&1 ) +s S. Otherwise s is a non-coset element of S. Now we divide the generating subsets of Z p a into three parts: A generating subset S Z p a "[0] is said to be I-type if S contains a non-coset element s with p |% s, III-type if S has a p r-partition (0
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2. PRELIMINARIES In this section, we shall quote and give some preliminary results for the purpose of characterizing the DCI-subsets of Z p a . The first result due to Babai [5] is a criterion for a subset of a finite group to be a DCI-subset. Lemma 2.1 (Babai [5]). Let C(G, S) be a Cayley graph of finite group G with respect to S G"[1], and A=Aut(C(G, S)). Then S is a CI-subset of G, if and only if, for any { # Sym(G) (the symmetric group on G) with R(G) { A, there exists an : # A such that R(G) { =R(G) :. Let A=Aut(C(G, S)), N A (R(G))=[: # A | R(G) : =R(G)] the ``normolizer'' of R(G) in A and Aut(G, S)=[: # Aut(G) | S : =S]. The next result due to C. D. Godsil [6] is helpful in the use of Babai's criterion. Lemma 2.2 (Godsil [6]). Let C(G, S) be a Cayley graph of finite group G with respect to S G "[1] and A=Aut(C(G, S)). Then N A (R(G))= R(G) < Aut(G, S), where < denotes the semi-direct product. It is generally known that the above two results are also valid for Cayley digraphs, which means that Babai's criterion is suitable to the judgment of DCI-subset of G. The following result is a consequence of Proposition 4.1 in [16], the conclusion of which is drawn for CI-subsets. Here, for the integrity, we prefer to give its proof for DCI-subsets. Lemma 2.3. Let C(G, S) be a Cayley digraph of finite p-group G with respect to S G "[1]. If p |% |Aut(G, S)|, then S is a DCI-subset of G. Proof. Let A=Aut(C(G, S)). If R(G) is a Sylow p-subgroup of A, then it immediately gets our result by Sylow Theorem and Lemma 2.1. Otherwise suppose P is the Sylow p-subgroup of A with P>R(G). Then by Lemma 2.2 R(G)
CLASSIFICATION OF (DCI )(CI )-SUBSETS
27
Proof. Let G*=.R(G) . &1. Then G* A=Aut(C(G, S)). Let (u, v) be any arc of C(G, S 1 ), that is, vu &1 # S 1 . It is easy to check that :=R(u) &1 .R((u . ) &1 ) . &1 # A 1 . We have (vu &1 ) : # S 1 , i.e., ((vu &1u) . (u . ) &1 ) . # S 1 , that is v . (u . ) &1 # S .1 =T 1 . Hence . is an isomorphism from C(G, S 1 ) to C(G, T 1 ). Similarly one can prove that . is an isomorphism from C(G, S 2 ) to C(G, T 2 ).
3. ISOMORPHISM OF CIRCULANT DIGRAPH OF LEXICOGRAPHIC PRODUCT Let 1 1 =(V 1 , E 1 ) and 1 2 =(V 2 , E 2 ) be two digraphs(or graphs). The lexicographic product of 1 1 by 1 2 is a digraph(or graph) 1=1 1[1 2 ] with vertex set V 1 _V 2 and ((u 1 , v 1 ), (u 2 , v 2 )) is an are (or an edge) if and only if either (u 1 , u 2 ) # E 1 or u 1 =u 2 and (v 1 , v 2 ) # E 2 . Let S Z n "[0]. Suppose S has a m-partition S=S 1 _ S 2 . Then by definition S 1 = . (( m) +s 1 )
and
S 2 ( m).
s1 # S1
It is easy to check that C(Z n , S) is the lexicographic product of C(Z m , S 1 ) by C(Z nm , S 2 ), where S 1 =S 1 ( m) Z n (m) =Z m and S 2 =[s 2 m | s 2 # S 2 ] Z nm . Recall that S generates Z n therefore S 1 {<. If S 2 =<, then C(G, S)=C(Z m , S 1 )[K cnm ], where K cnm is the complement of the complete graph K nm , which will be called the coset circulant digraph. A m-partition S=S 1 _ S 2 is said to be maximal if there is no m$ | m (1
(1)
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HUANG AND MENG
In fact, if |S 1 | =1 then |N + 1 (x) & S| = |S 2 | < |S 1 | for any x # S 1 . We now suppose |S 1 | 2 in what follows. It is easy to see that there exists v # S 1 such that v adjoins to at most |S 1 | &2 vertices of S 1 in 1 1 , since otherwise 1 1 will be a complete graph. Hence |N + 1 | |S 1 | &2. This implies 11(v ) & S that for any x # (m) +v S 1 (where v =( m) +v) |N + 1 (x) & S|
n n . ( |S 1 | &2)+2 |S 2 | = |S 1 | +2 |S 2 | & m m
\
+
Note that S 2 /( m), we have |N + 1 (x) & S| < |S 1 |. On other aspect, for any y # S 2 /( m) |N + 1 ( y) & S| |S 1 |.
(2)
Let 2 1 , 2 2 , ..., 2 d be some orbits of A 0 acting on S, each of which contains at least an element of S 2 and S 2 1 i d 2 i /S. Since C(Z n , S) is the lexicographic product of C(Z m , S 1 ) by C(Z nm , S 2 ), the action of A 0 must be transitive on ( m) +s 1 for s 1 # S 1 . This means that ( m) +s 1 2 i if 2 i & (( m) +s 1 ){< (1 i d ). "( 1 i d 2 i ). Then S * Let S *=S 1 1 is the union of some cosets of ( m), and by (1) and (2), S * {<. Additionally, we have S*1 { =S * and 1 1 { ( 1 i d 2 i ) = 1 i d 2 i for { # A 0 . Now we consider 1 *=C(Z which is clearly a coset circulant 1 n , S *), 1 digraph. Set K*=[u # Z n | u+S 1*=S 1*]. It is simple to show that K* is a subgroup of Z n and ( m) K*. Notice that the vertices of K* are just all the vertices which are out adjacent in C(Z n , S *). We have K* { =K* for to every vertex of S * 1 1 : # Aut(C(Z n , S *)) | 0 =0]. By Lemma 2.4, A 0 A 0* , and hence { # A *=[: 0 1 K* : =K* for : # A 0 . Since 2 i is an orbit of A 0 and ( m) K*, we deduce is a m$-partithat 2 i K* (i=1, 2, ..., d ). It implies that S=S * 1 _ (S "S *) 1 tion, where K*=( m$). Since S=S 1 _ S 2 is maximal, we claim that K*=( m). Hence, for any { # A 0 , S {2 =S 2 and so S {1 =S 1 . We thus complete this proof. Lemma 3.2. Let S Z n "[0]. If S=S 1 _ S 2 is maximal m-partition, then C(Z n , S)$C(Z n , T ) if and only if there is maximal m-partition T=T 1 _ T 2 such that C(Z n , S 1 )$C(Z n , T 1 ) and C(Z n , S 2 )$C(Z n , T 2 ). Proof. Since C(Z n , S) is the lexicographic product of C(Z m , S 1 ) by C(Z nm , S 2 ), where S 1 =S 1 (m) Z n ( m) =Z m and S 2 =[s 2 m | s 2 # S 2 ] Z nm , the sufficiency is evident. Now we prove the necessity.
CLASSIFICATION OF (DCI )(CI )-SUBSETS
29
If C(Z n , S) is a complete graph, then C(Z n , T ) is also a complete graph, and hence our result is valid. If S 2 /( m) and S 1 =Z n "( m), then C(Z m , S 1 ) is a complete graph. Let S c =Z n "(S _ [0]) and T c =Z n "(T _ [0]). Then C(Z n , S c ) and C(Z n , T c ) are the complements of C(Z n , S) and C(Z n , T ) respectively. Since C(Z n , S)$C(Z n , T ), C(Z n , S c )$C(Z n , T c ). Clearly C(Z n , S c ) is several copy of C(( S c ), S c ). Hence C(( S c ), S c )$C(( T c ), T c ).
(3)
Notice that Z n is a cyclic group, ( T c ) =( S c ) ( m). This implies that T has maximal m-partition: T=T 1 _ T 2 with T 1 =Z n "( m) and T 2 ( m). From (3), C(Z n , S 2 )$C(Z n , T 2 ). In addition, both C(Z m , S 1 ) and C(Z m , T 1 ) (T 1 =T 1 ( m) ) are complete graphs, C(Z n , S 1 )$C(Z n , T 1 ). At last we suppose S 1 {Z n "( m). Let . be an isomorphism from C(Z n , S) to C(Z n , T ) with 0 . =0. Set T 1 =S .1 and T 2 =S .2 . Then by Lemma 3.1 and Lemma 2.4, C(Z n , S 1 )$C(Z n , T 1 )
and
C(Z n , S 2 )$C(Z n , T 2 ).
For any isomorphism f from C(Z n , S) to C(Z n , T ) with 0 f =0, &1 =S 1 .f &1 # A 0 =[: # Aut(C(Z n , S)) | 0 : =0]. Again by Lemma 3.1, S .f 1 and S 1f =S .1 =T 1 . Let u # (m) and f =R(u) .R(&u . ). Then f is an isomorphism from C(Z n , S) to C(Z n , T ) with 0 f =0. Thus we have .
.R(&u ) =S 1f =T 1 O (S 1 +u) . =T 1 +u . O S .1 =T 1 +u . O S R(u) 1
T 1 =T 1 +u .
for any
u # ( m).
(4)
Let K=[v # Z n | v+T 1 =T 1 ]. Then K is a subgroup of Z n and T 1 = t1 # T1 (K+t 1 ). By (4), ( m) . K. Note that each vertex of ( m) is out-adjacent to every vertex of S 1 in C(Z n , S 1 ) and each vertex of K is outadjacent to every vertex of T 1 in C(Z n , T 1 ). Additionally, m-partition of S=S 1 _ S 2 is maximal, there is no other vertex which is out-adjacent to every vertex of S 1 in C(Z n , S 1 ) except that in ( m). We deduce ( m) . =K, i.e., (m) . =( m). It then follows T 1 = t1 # T1 (( m) +t 1 ) and T 2 =S .2 ( m). Hence T=T 1 _ T 2 is a m-partition which is clearly maximal. We complete the proof. It is well known that Aut(Z n )=[* # Z n | gcd(*, n)=1] and * # Aut(Z n ) is defined by *: u Ä *u(mod n),
for
\u # Z n .
Hence Aut(Z n , S)=[* # Aut(Z n ) | *S=S] Aut(Z n ).
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Theorem 3.1. Let S Z n "[0]. If S=S 1 _ S 2 is maximal m-partition, then S is a DCI-subset of Z n if and only if both S 1 and S 2 are DCI-subsets of Z n and (Aut(Z n , S 1 ), Aut(Z n , S 2 )) =Aut(Z n ). Proof. If S=Z n "[0], C(Z n , S) is a complete graph, and our result clearly holds. In the following we assume S/Z n "[0]. We first prove the sufficiency. Let C(Z n , S)$C(Z n , T ) and . the isomorphism from C(Z n , S) to C(Z n , T ) with 0 . =0. Set T 1 =S .1 and T 2 =S .2 . By Lemma 3.2, T=T 1 _ T 2 is maximal m-partition, C(Z n , S 1 )$ C(Z n , T 1 ) and C(Z n , S 2 )$C(Z n , T 2 ). According to assumption that S 1 and S 2 are DCI-subsets of Z n , there are * 1 and * 2 in Aut(Z n ) such that T 1 =* 1 S 1 and T 2 =* 2 S 2 . Since ( Aut(Z n , S 1 ), Aut(Z n , S 2 )) =Aut(Z n ), there exists ' i # Aut(Z n , S 1 ) (i=1, 2, ..., d) such that Aut(Z n )=' 1 Aut(Z n , S 2 ) _ ' 2 Aut(Z n , S 2 ) _ } } } _ ' d Aut(Z n , S 2 ). Thus there is ' i (1 i d ) such that ' i Aut(Z n , S 2 )=* &1 1 * 2 Aut(Z n , S 2 ) O Aut(Z n , S 1 ) & * &1 1 * 2 Aut(Z n , S 2 ){<. Hence there is * # Aut(Z n ) such that * # * 1 Aut(Z n , S 1 ) & * 2 Aut(Z n , S 2 ). It follows that *S 1 =* 1 S 1 =T 1 and *S 2 =* 2 S 2 =T 2 , that is, T=*S and so S is a DCI-subset of Z n . Conversely, let S/Z n "[0] be a DCI-subset of Z n and S=S 1 _ S 2 maximal m-partition. Suppose C(Z n , S 1 )$C(Z n , T 1 )
and
C(Z n , S 2 )$C(Z n , T 2 ).
Set T=T 1 _ T 2 , which is clearly maximal m-partition. Then by Lemma 3.2, C(Z n , S)$C(Z n , T ), and hence there is * # Aut(Z n ) such that T=*S. It follows that T 1 =*S 1 and T 2 =*S 2 . Therefore S 1 and S 2 are DCI-subsets of Z n . In addition, taking T=T 1 _ T 2 , where T 1 =* 1 S 1 and T 2 =* 2 S 2 (* 1 , * 2 # Aut(Z n )), similarly we get C(Z n , S)$C(Z n , T ), and hence there is * # Aut(Z n ) such that T=*S, i.e., *S 1 =* 1 S 1 and *S 2 =* 2 S 2 . So we have * # * 1 Aut(Z n , S 1 ) & * 2 Aut(Z n , S 2 ), that is, Aut(Z n , S 1 ) & * &1 1 * 2 Aut(Z n , S 2 ){<.
(5)
Let + # Aut(Z n ) be arbitrary. By (5), Aut(Z n , S 1 ) & + Aut(Z n , S 2 ){<. Hence there exist + 1 # Aut(Z n , S 1 ) and + 2 # Aut(Z n , S 2 ) such that + 1 =++ 2 , so +=+ 1 + &1 2 . It completes the proof.
CLASSIFICATION OF (DCI )(CI )-SUBSETS
31
4. CLASSIFICATION OF DCI-SUBSETS OF Z p a Let p be an odd prime. In this section we shall completely characterize the DCI-subsets of Z p a . It is well known that Aut(Z n ) is an abelian group of order .(n), where . is the Euler function. In particular, for n= p a we have the following. Lemma 4.1. Let p be an odd prime. Then Aut(Z p a ) is a cyclic group of order p a&1 ( p&1) and ( p i ) +1 is the unique subgroup of order p a&i (0
(6)
Then 1 k
and
S 1 =S "S 2 .
(7)
By definition each s 1 # S 1 is a coset element. We call j the coset index of s 1 in S 1 if ( p j ) +s 1 S 1 and ( p j&1 ) +s 1 3 S 1 . It is clear 1 j a&1. Set t=max[t | j is the coset index of s 1 and s 1 # S 1 ].
(8)
By definition ( p t ) +s 1 S 1 for s 1 # S 1 and hence S 1 = s1 # S1 (( p t ) +s 1 ). Additionally k
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HUANG AND MENG
Lemma 4.2. Let S be a II-type subset of Z p a (a 3) and S=S 1 _ S 2 is defined in (7). Set T=T 1 _ T 2 , where T 2 =S 2 and T 1 =*S 1 for * # ( p t&k ) +1 (k and t are defined in (6) and (8)). Then C(Z p a , S)$ C(Z p a , T ). Proof. Let u=x 1 p k + y 1 # Z p a * # ( p t&k ) +1, let us define f * as
(0 x 1 < p a&k, 0 y 1 < p k ).
For
f * : u Ä u f* =x 1 p k +*y 1 . It is routine to verify that f * is an one to one mapping on Z p a . Let v=x 2 p k + y 2 # Z p a (0 x 2 < p a&k, 0 y 2 < p k ). Then v f* &u f* =(x 2 &x 1 ) p k +*( y 2 & y 1 )=(x 2 &x 1 ) p k (1&*)+*(v&u).
(9)
Suppose (u, v) is an arc of C(Z p a , S). It suffices to show that (u f*, v f* ) is an arc of C(Z p a , T ). If v&u=(x 2 &x 1 ) p k +( y 2 & y 1 ) # S 2 , y 1 = y 2 and hence v f* &u f* = (x 2 &x 1 ) p k =v&u # S 2 =T 2 . Suppose v&u # S 1 . Let j be the coset index of v&u. Then ( p j ) +(v&u) S 1 O ( p j ) +*(v&u) *S 1 =T 1 .
(10)
By definition of t in (8), t j and ( p t ) ( p j ). Since * # ( p t&k ) +1, 1&* # ( p t&k ) and hence (x 2 &x 1 ) p k (1&*) # ( p t ) ( p j ). By (9) and (10), we have v f* &u f* # ( p j ) +*(v&u) T 1 . It completes the proof. Theorem 4.2. Let S be a II-type subset of Z p a ( 3). Then S is a NDCI-subset of Z p a (a 3) (i.e., S is not a DCI-subset). Proof. Let S=S 1 _ S 2 be a II-type subset of Z p a , which is defined in (7). Let T=T 1 _ T 2 , where T 1 =*S 1 (* # ( p t&k ) +1) and T 2 =S 2 . By Lemma 4.2 C(Z p a , S)$C(Z p a , T ). To the contrary suppose S is a DCI-subset. Then there is ' # Aut(Z p a ) such that T='S, that is, 'S 1 =*S 1 and 'S 2 =S 2 O ' # * Aut(Z p a , S 1 ) & Aut(Z p a , S 2 ). It implies for any * # ( p t&k ) +1 * Aut(Z p a , S 1 ) & Aut(Z p a , S 2 ){<.
(11)
Remember that S 2 contains a non-coset element s* with p k & s*. We claim that ( p a&k&1 ) +1 Aut(Z p a , S 2 ).
(12)
Since otherwise (( p a&k&1 ) +1) s*=( p a&1 ) +s* S 2 , which contradicts the assumption of s*.
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CLASSIFICATION OF (DCI )(CI )-SUBSETS
Since k
(13)
According to the definition of t in (8), there exists s 1* # S 1 such that ( p t ) +s 1* S 1 ,
but
( p t&1 ) +s 1* 3 S 1 .
(14)
Let p l & |Aut(Z p a , S 1 )|. Then ( p a&l ) +1 Aut(Z p a , S 1 ). Suppose p j & s 1* . It is clear 0 j
(15)
Let H$=( Aut(Z p a , S 1 ), Aut(Z p a , S 2 )) and H=( Aut(Z p a , S 1 ), Aut(Z p a , S2 ), ( p t&k ) +1). Then H$, H Aut(Z p a ) and by Lemma 4.1, (13), and (15) we claim that ( p t&k ) +1 H$. Hence H$
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ACKNOWLEDGMENTS The author thanks the referees for their insightful comments and helpful suggestions, which have simplified the proof of Lemma 2.4 and also improved the presentation of this paper.
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