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On regular association schemes of order pq
Discrete Mathematics 338 (2015) 111–113
Contents lists available at ScienceDirect
Discrete Mathematics journal homepage: www.elsevier.com/locate/disc
Note
On regular association schemes of order pq Masayoshi Yoshikawa Nagano Prefecture Azusagawa Senior High School, 10000-1 Hata, Matsumoto, Nagano, 390-1401, Japan
article
info
Article history: Received 4 June 2014 Received in revised form 28 August 2014 Accepted 30 August 2014
abstract We show that each non-thin regular association scheme of order pq (p and q two not necessarily different prime numbers) is a wreath product of two thin cyclic schemes of order p and q. © 2014 Elsevier B.V. All rights reserved.
Keywords: Association schemes Girth Regular Strong girth Wreath product
1. Introduction In [2] and [3], the author started investigating association schemes all relations s of which satisfy s∗ ss = {s}. (Such schemes are called regular.) In [3, Theorem 19], it was shown that regular schemes S always have a normal thin closed subset of prime valency, and in [3, Theorem 20], it was shown that the quotient scheme of S over this thin closed subset is again regular if S is commutative. This indicates that regular association schemes may play an interesting role in the extension theory of association schemes. The regularity condition says that s∗ ss = {s} for each relation s in S. Since regular relations satisfy sg(s) < ∞ from Lemma 7, they satisfy ss∗ = s∗ s from Lemma 3. As a consequence, they satisfy ss∗ s = {s}. This condition has been investigated by Zieschang in [4]. From the first of the above two results [3, Theorems 19 and 20] one obtains of course that each regular association scheme of prime order must be thin. In the present paper, we prove that each regular association scheme of order pq (p and q two not necessarily different prime numbers) is thin or the wreath product of two thin cyclic schemes of order p and q. 2. Preliminaries For association schemes, refer to [5]. Let X be a finite set and S be a partition of X × X which does not contain the empty set. For s ∈ S, we define the adjacency matrix σs of s by the matrix whose both rows and columns are indexed by X and the (x, y)-entry is 1 if (x, y) ∈ s and 0 otherwise. The set S is called an association scheme on the set X if 1. 1 := {(x, x) | x ∈ X } ∈ S, 2. if s ∈ S, then s∗ := {(y, x) | (x, y) ∈ s} ∈ S, 3. for s, t , u ∈ S, there exists an integer astu such that σs σt = u∈S astu σu .
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.disc.2014.08.027 0012-365X/© 2014 Elsevier B.V. All rights reserved.
112
M. Yoshikawa / Discrete Mathematics 338 (2015) 111–113
For s ∈ S, we define ns := ass∗ 1 and call it the valency of s. For C ⊂ S, we put σC := c ∈C σc and nC := c ∈C nc . The integer nC is called the valency of C if C is a closed subset of S. We denote the quotient scheme of S over a closed subset C by S //C . Let s ∈ S, let n ∈ N \ {0, 1, 2}, and let f1 , . . . , fn ∈ S be given. Then af1 ···fn ℓ is defined recursively by
af1 ···fn s =
af1 ···fn−1 ℓ aℓfn s .
ℓ∈S
For any relations s, ℓ, ℓ′ ∈ S and n ≥ 2, we set asn ℓ = as · · · s ℓ and aℓ′ sn ℓ = aℓ′ s · · · s ℓ .
n
n
We define the girth and the strong girth for association schemes. The rest of this section overlaps with some of the contents in [2] and [3]. However the author believes that it helps the readers understand regular association schemes. Definition 1. We define the girth g(s) of a non-identity relation s of S by g(s) = min{e ∈ N \ {0, 1} | ase 1 ̸= 0}. It is well-known that g(s) is a positive integer for any s ∈ S. We define the strong girth sg(s) of a non-identity relation s of S by sg(s) = min{e ∈ N \ {0, 1} | ase 1 = nes −1 }, and if there exist no such natural numbers, we define that sg(s) = ∞. We define that g(1) = sg(1) = 1. We introduce some results on relations with finite strong girth. Proposition 2 ([2]). Let s ∈ S be a relation with e = sg(s) < ∞. If e ≥ 3, it follows that 1. ase 1 = ns e−1 . 2. ase−1 s∗ = ns e−2 . 3. ase−1 ℓ = 0 for ℓ ̸= s∗ . Lemma 3. If sg(s) < ∞, the followings hold. 1. 2.
s∗ s = ss∗ , s∗ s is a closed subset of S.
Proof. We set e = sg(s). Namely, we know that σs e−1 = ns e−2 σs∗ and se−1 = {s∗ }. Thus ss∗ = se = s∗ s and (1) holds. We have that (s∗ s)(s∗ s)∗ = s∗ ss∗ s = s∗ sss∗ = ss∗ = s∗ s and (2) holds. Generally it follows that g(s) ≤ sg(s) for any relation s. It is easy to find some association schemes which have a relation with g(s) ̸= sg(s). For example, a non-identity relation s of as07 [2] in the classification [1] by Hanaki and Miyamoto has g(s) = 3 but sg(s) = ∞. It is shown in [2] that sg(s) = g(s) if sg(s) < ∞. Theorem 4 ([2]). Let s ∈ S be a relation with sg(s) < ∞. Then, sg(s) = g(s). Corollary 5 ([3]). Let s ∈ S be a relation of S. If there exists a number t such that ast 1 = nts−1 , it follows that sg(s) | t. The converse of this corollary does not hold generally (for example, the non-identity relation of a rank two association scheme of order |X | > 2). We will consider a condition providing that the converse of this corollary holds. Let s be a relation of finite strong girth e. Namely we have ase 1 = ns e−1 and ase−1 ℓ = δℓs∗ ns e−2 for any ℓ ∈ S. By the direct calculation, it follows that as2e 1 =
ℓ∈S
ase−1 ℓ aℓse+1 1 = ns e−2 as∗ se+1 1
= ns e−2
= ns e−1
ℓ∈S ℓ∈S
as∗ ℓ1 ase+1 ℓ = ns e−1 ase+1 s ase−1 ℓ aℓs2 s = ns 2e−3 as∗ sss .
Therefore the converse of Corollary 5 holds for t = 2sg(s) if as∗ sss = ns 2 . Proposition 6 ([3]). Let s be a relation. The followings are equivalent. 1. 2. 3.
as∗ sss = ns 2 , σs∗ σs σs = ns 2 σs , s∗ ss = {s},
Proof. It is clear from the definition.
If a relation s satisfies one of three conditions, we say that the relation s is regular. We investigate regular relations.
M. Yoshikawa / Discrete Mathematics 338 (2015) 111–113
113
Lemma 7 ([3]). If a relation s is regular, sg(s) < ∞. The following theorem says that the converse of Corollary 5 holds for regular relations. Theorem 8 ([3]). Let s be a regular relation. We have that
ast 1 =
ns t − 1 0
if sg(s)|t , otherwise.
From Lemma 7, we know that if a relation s is regular, its strong girth sg(s) < ∞. However, the converse of this lemma does not always hold. For example, the strong girth of the non-identity relation of any rank two association scheme of order |X | > 2 is 2, but the relation is not regular. We will prove that counterexamples of the converse of this lemma are only symmetric relations. Theorem 9 ([3]). Let s be a relation of finite strong girth e ̸= 2. The relation s is regular. An association scheme all relations of which are regular is called a regular association scheme. Any regular association scheme has the following important property. Theorem 10 ([3]). Let S be a regular association scheme. Then, S has a non-identity relation of valency 1. Corollary 11. Let p be a prime. A regular association scheme of order p is thin. Remark 12. Any thin association scheme is regular. 3. Regular association schemes of order pq In this section, we consider non-thin regular association schemes of order pq for primes p and q. Let p and q be primes and S be a non-thin regular association scheme of order pq. Due to Theorem 10, we may assume that S has a thin closed subset C of valency p. Lemma 13. Let s ∈ S \ C be given. Then, ns = p. Proof. First we assume that p ̸= q. Since s is regular, ns = ns∗ s and s∗ s is a closed subset of S. Thus it follows that ns = 1, p, or q. If ns = 1, we know that ⟨C , s⟩ = S and S is thin. If ns = q, ns∗ s = q and s∗ s is a thin closed subset of valency q. Thus S is thin. Therefore we have that ns = p. If p = q, we obtain ns ̸= 1 in a similar way as we obtained it in the case where p ̸= q. This leads to ns = p. We know that the order of the quotient scheme S //C is q. Theorem 14. The quotient scheme S //C is thin. Proof. Let s ∈ S \ C be given. From Lemma 13, it follows that ns = ns∗ s = p and s∗ s is a closed subset of S. Thus s∗ s = C because each relation in S \ C has valency p. Since s is regular, we have s∗ ss = {s} and s∗ s = ss∗ . Thus, CsC = s∗ ssC = sC = ss∗ s = s∗ ss = {s} and nCsC = ns = p. From [5, Theorem 4.1.3 (iii)] we know that nsC nC = nCsC . Thus nsC = 1. From the above argument, a non-thin regular association scheme of order pq has a thin closed subset C of valency p and the quotient scheme S //C of S by C is isomorphic to the thin cyclic scheme of order q. The following theorem is our goal in this paper. Theorem 15. Let p and q be primes. A non-thin regular association scheme of order pq is the wreath product of two thin cyclic schemes of order p and q. Proof. We assume that S has a thin closed subset C of valency p. From Lemma 13, it follows that |S \ C | = q − 1. Moreover we have that S //C is isomorphic to the thin cyclic scheme of order q from Theorem 14. This fact means that S is the wreath product of two thin cyclic schemes of order p and q. Acknowledgment The author would like to thank the anonymous referee for his/her carefully reading the article and giving valuable comments. References [1] [2] [3] [4] [5]
A. Hanaki, I. Miyamoto, The classification of association schemes with small vertices, http://kissme.shinshu-u.ac.jp/as/. M. Yoshikawa, On association schemes of exponent three, preprint. M. Yoshikawa, On association schemes of finite exponent, preprint. P.-H. Zieschang, Association schemes in which the thin residue is a finite cyclic group, J. Alg 324 (12) (2010) 3572–3578. P.-H. Zieschang, Theory of Association Schemes, Springer Monographs in Mathematics, Berlin, Heidelberg, New York, 2005.
Contents lists available at ScienceDirect
Discrete Mathematics journal homepage: www.elsevier.com/locate/disc
Note
On regular association schemes of order pq Masayoshi Yoshikawa Nagano Prefecture Azusagawa Senior High School, 10000-1 Hata, Matsumoto, Nagano, 390-1401, Japan
article
info
Article history: Received 4 June 2014 Received in revised form 28 August 2014 Accepted 30 August 2014
abstract We show that each non-thin regular association scheme of order pq (p and q two not necessarily different prime numbers) is a wreath product of two thin cyclic schemes of order p and q. © 2014 Elsevier B.V. All rights reserved.
Keywords: Association schemes Girth Regular Strong girth Wreath product
1. Introduction In [2] and [3], the author started investigating association schemes all relations s of which satisfy s∗ ss = {s}. (Such schemes are called regular.) In [3, Theorem 19], it was shown that regular schemes S always have a normal thin closed subset of prime valency, and in [3, Theorem 20], it was shown that the quotient scheme of S over this thin closed subset is again regular if S is commutative. This indicates that regular association schemes may play an interesting role in the extension theory of association schemes. The regularity condition says that s∗ ss = {s} for each relation s in S. Since regular relations satisfy sg(s) < ∞ from Lemma 7, they satisfy ss∗ = s∗ s from Lemma 3. As a consequence, they satisfy ss∗ s = {s}. This condition has been investigated by Zieschang in [4]. From the first of the above two results [3, Theorems 19 and 20] one obtains of course that each regular association scheme of prime order must be thin. In the present paper, we prove that each regular association scheme of order pq (p and q two not necessarily different prime numbers) is thin or the wreath product of two thin cyclic schemes of order p and q. 2. Preliminaries For association schemes, refer to [5]. Let X be a finite set and S be a partition of X × X which does not contain the empty set. For s ∈ S, we define the adjacency matrix σs of s by the matrix whose both rows and columns are indexed by X and the (x, y)-entry is 1 if (x, y) ∈ s and 0 otherwise. The set S is called an association scheme on the set X if 1. 1 := {(x, x) | x ∈ X } ∈ S, 2. if s ∈ S, then s∗ := {(y, x) | (x, y) ∈ s} ∈ S, 3. for s, t , u ∈ S, there exists an integer astu such that σs σt = u∈S astu σu .
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.disc.2014.08.027 0012-365X/© 2014 Elsevier B.V. All rights reserved.
112
M. Yoshikawa / Discrete Mathematics 338 (2015) 111–113
For s ∈ S, we define ns := ass∗ 1 and call it the valency of s. For C ⊂ S, we put σC := c ∈C σc and nC := c ∈C nc . The integer nC is called the valency of C if C is a closed subset of S. We denote the quotient scheme of S over a closed subset C by S //C . Let s ∈ S, let n ∈ N \ {0, 1, 2}, and let f1 , . . . , fn ∈ S be given. Then af1 ···fn ℓ is defined recursively by
af1 ···fn s =
af1 ···fn−1 ℓ aℓfn s .
ℓ∈S
For any relations s, ℓ, ℓ′ ∈ S and n ≥ 2, we set asn ℓ = as · · · s ℓ and aℓ′ sn ℓ = aℓ′ s · · · s ℓ .
n
n
We define the girth and the strong girth for association schemes. The rest of this section overlaps with some of the contents in [2] and [3]. However the author believes that it helps the readers understand regular association schemes. Definition 1. We define the girth g(s) of a non-identity relation s of S by g(s) = min{e ∈ N \ {0, 1} | ase 1 ̸= 0}. It is well-known that g(s) is a positive integer for any s ∈ S. We define the strong girth sg(s) of a non-identity relation s of S by sg(s) = min{e ∈ N \ {0, 1} | ase 1 = nes −1 }, and if there exist no such natural numbers, we define that sg(s) = ∞. We define that g(1) = sg(1) = 1. We introduce some results on relations with finite strong girth. Proposition 2 ([2]). Let s ∈ S be a relation with e = sg(s) < ∞. If e ≥ 3, it follows that 1. ase 1 = ns e−1 . 2. ase−1 s∗ = ns e−2 . 3. ase−1 ℓ = 0 for ℓ ̸= s∗ . Lemma 3. If sg(s) < ∞, the followings hold. 1. 2.
s∗ s = ss∗ , s∗ s is a closed subset of S.
Proof. We set e = sg(s). Namely, we know that σs e−1 = ns e−2 σs∗ and se−1 = {s∗ }. Thus ss∗ = se = s∗ s and (1) holds. We have that (s∗ s)(s∗ s)∗ = s∗ ss∗ s = s∗ sss∗ = ss∗ = s∗ s and (2) holds. Generally it follows that g(s) ≤ sg(s) for any relation s. It is easy to find some association schemes which have a relation with g(s) ̸= sg(s). For example, a non-identity relation s of as07 [2] in the classification [1] by Hanaki and Miyamoto has g(s) = 3 but sg(s) = ∞. It is shown in [2] that sg(s) = g(s) if sg(s) < ∞. Theorem 4 ([2]). Let s ∈ S be a relation with sg(s) < ∞. Then, sg(s) = g(s). Corollary 5 ([3]). Let s ∈ S be a relation of S. If there exists a number t such that ast 1 = nts−1 , it follows that sg(s) | t. The converse of this corollary does not hold generally (for example, the non-identity relation of a rank two association scheme of order |X | > 2). We will consider a condition providing that the converse of this corollary holds. Let s be a relation of finite strong girth e. Namely we have ase 1 = ns e−1 and ase−1 ℓ = δℓs∗ ns e−2 for any ℓ ∈ S. By the direct calculation, it follows that as2e 1 =
ℓ∈S
ase−1 ℓ aℓse+1 1 = ns e−2 as∗ se+1 1
= ns e−2
= ns e−1
ℓ∈S ℓ∈S
as∗ ℓ1 ase+1 ℓ = ns e−1 ase+1 s ase−1 ℓ aℓs2 s = ns 2e−3 as∗ sss .
Therefore the converse of Corollary 5 holds for t = 2sg(s) if as∗ sss = ns 2 . Proposition 6 ([3]). Let s be a relation. The followings are equivalent. 1. 2. 3.
as∗ sss = ns 2 , σs∗ σs σs = ns 2 σs , s∗ ss = {s},
Proof. It is clear from the definition.
If a relation s satisfies one of three conditions, we say that the relation s is regular. We investigate regular relations.
M. Yoshikawa / Discrete Mathematics 338 (2015) 111–113
113
Lemma 7 ([3]). If a relation s is regular, sg(s) < ∞. The following theorem says that the converse of Corollary 5 holds for regular relations. Theorem 8 ([3]). Let s be a regular relation. We have that
ast 1 =
ns t − 1 0
if sg(s)|t , otherwise.
From Lemma 7, we know that if a relation s is regular, its strong girth sg(s) < ∞. However, the converse of this lemma does not always hold. For example, the strong girth of the non-identity relation of any rank two association scheme of order |X | > 2 is 2, but the relation is not regular. We will prove that counterexamples of the converse of this lemma are only symmetric relations. Theorem 9 ([3]). Let s be a relation of finite strong girth e ̸= 2. The relation s is regular. An association scheme all relations of which are regular is called a regular association scheme. Any regular association scheme has the following important property. Theorem 10 ([3]). Let S be a regular association scheme. Then, S has a non-identity relation of valency 1. Corollary 11. Let p be a prime. A regular association scheme of order p is thin. Remark 12. Any thin association scheme is regular. 3. Regular association schemes of order pq In this section, we consider non-thin regular association schemes of order pq for primes p and q. Let p and q be primes and S be a non-thin regular association scheme of order pq. Due to Theorem 10, we may assume that S has a thin closed subset C of valency p. Lemma 13. Let s ∈ S \ C be given. Then, ns = p. Proof. First we assume that p ̸= q. Since s is regular, ns = ns∗ s and s∗ s is a closed subset of S. Thus it follows that ns = 1, p, or q. If ns = 1, we know that ⟨C , s⟩ = S and S is thin. If ns = q, ns∗ s = q and s∗ s is a thin closed subset of valency q. Thus S is thin. Therefore we have that ns = p. If p = q, we obtain ns ̸= 1 in a similar way as we obtained it in the case where p ̸= q. This leads to ns = p. We know that the order of the quotient scheme S //C is q. Theorem 14. The quotient scheme S //C is thin. Proof. Let s ∈ S \ C be given. From Lemma 13, it follows that ns = ns∗ s = p and s∗ s is a closed subset of S. Thus s∗ s = C because each relation in S \ C has valency p. Since s is regular, we have s∗ ss = {s} and s∗ s = ss∗ . Thus, CsC = s∗ ssC = sC = ss∗ s = s∗ ss = {s} and nCsC = ns = p. From [5, Theorem 4.1.3 (iii)] we know that nsC nC = nCsC . Thus nsC = 1. From the above argument, a non-thin regular association scheme of order pq has a thin closed subset C of valency p and the quotient scheme S //C of S by C is isomorphic to the thin cyclic scheme of order q. The following theorem is our goal in this paper. Theorem 15. Let p and q be primes. A non-thin regular association scheme of order pq is the wreath product of two thin cyclic schemes of order p and q. Proof. We assume that S has a thin closed subset C of valency p. From Lemma 13, it follows that |S \ C | = q − 1. Moreover we have that S //C is isomorphic to the thin cyclic scheme of order q from Theorem 14. This fact means that S is the wreath product of two thin cyclic schemes of order p and q. Acknowledgment The author would like to thank the anonymous referee for his/her carefully reading the article and giving valuable comments. References [1] [2] [3] [4] [5]
A. Hanaki, I. Miyamoto, The classification of association schemes with small vertices, http://kissme.shinshu-u.ac.jp/as/. M. Yoshikawa, On association schemes of exponent three, preprint. M. Yoshikawa, On association schemes of finite exponent, preprint. P.-H. Zieschang, Association schemes in which the thin residue is a finite cyclic group, J. Alg 324 (12) (2010) 3572–3578. P.-H. Zieschang, Theory of Association Schemes, Springer Monographs in Mathematics, Berlin, Heidelberg, New York, 2005.