the spectrum of 2-idempotent 3-quasigroups with conjugate invariant subgroups

The Spectrum of 2-Idempotent3-Quasigroups with ConjugateInvariant Subgroups

Lijun Ji,1 R. Wei21Department of Mathematics, Suzhou University, Suzhou 215006, China,E-mail: [email protected]

2Department of Computer Science, Lakehead University, Thunder Bay, ON,Canada P7B 5E1, E-mail: [email protected]

Received August 11, 2009; revised January 19, 2010

Published online 17 March 2010 in Wiley InterScience (www.interscience.wiley.com).DOI 10.1002/jcd.20255

Abstract: A ternary quasigroup (or 3-quasigroup) is a pair (N,q) where N is an n-setand q(x,y,z) is a ternary operation on N with unique solvability. A 3-quasigroup is called2-idempotent if it satisfies the generalized idempotent law: q(x,x,y)=q(x,y,x)=q(y,x,x)=y.

A conjugation of a 3-quasigroup, considered as an OA(3,4,n), (N,B), is a permutation of thecoordinate positions applied to the 4-tuples of B. The subgroup of conjugations under which(N,B) is invariant is called the conjugate invariant subgroup of (N,B). In this article, wedetermined the existence of 2-idempotent 3-quasigroups of order n, n≡7 or 11 (mod 12) andn≥11, with conjugate invariant subgroup consisting of a single cycle of length three. Thisresult completely determined the spectrum of 2-idempotent 3-quasigroups with conjugateinvariant subgroups. As a corollary, we proved that an overlarge set of Mendelsohn triplesystem of order n exists if and only if n≡0,1 (mod 3) and n �=6. q 2010 Wiley Periodicals, Inc.J Combin Designs 18: 292–304, 2010

Keywords: 3-quasigroup; orthogonal array; quadruple system; ordered design

1. INTRODUCTION

A ternary quasigroup (or 3-quasigroup) is a pair (N ,q) where N is an n-set andq(x, y, z) is a ternary operation on N with unique solvability; that is, in the equation

Contract grant sponsor: NSFC; Contract grant numbers: 10701060; 10831002 (to L. J.); Contract grant sponsors:Qing Lan Project of Jiangsu Province (to L. J.); NSERC; Contract grant number: 239135-06 (to R. W.).

Journal of Combinatorial Designsq 2010 Wiley Periodicals, Inc. 292

THE SPECTRUM OF 2-IDEMPOTENT 3-QUASIGROUPS 293

w=q〈x, y, z〉 if values for any three of x , y, z, w are given, the value of the remainingvariable is uniquely determined. Alternately, a 3-quasigroup of order n can be definedas an orthogonal array OA(3,4,n), (N ,B), with B a collection of ordered 4-tuples(x, y, z,q(x, y, z)) for all choices of x , y, z∈N . A 3-quasigroup is called 2-idempotentif it satisfies the generalized idempotent law:

q(x, x, y)=q(x, y, x)=q(y, x, x)= y.

A conjugation of an OA(3,4,n) (or a 3-quasigroup), (N ,B), is a permutation � ofthe coordinate positions applied to the 4-tuples of B by replacing each ordered 4-tuple(x1, x2, x3, x4) by (x�(1), x�(2), x�(3), x�(4)). The subgroup of conjugations under which(N ,B) is invariant is called the conjugate invariant subgroup of (N ,B). Clearly, theconjugate invariant subgroup of a 3-quasigroup is a subgroup of S4, where S4 is thesymmetric group on a 4-set.A 2-idempotent 3-quasigroup is related to an ordered design OD(3,4,n). An ordered

design OD(t,k,n) can be defined as a k-N array (an array over N with k columns) A,where |N |=n, satisfying the properties that each row of A contains k distinct elements ofN and every t columns of A contain each ordered t-subset of N exactly once. Similarly,the subgroup of conjugations under which a k-N array A is invariant is called theconjugate invariant subgroup of A. If an OD(t,k,n) has a conjugate invariant subgroupG, then it is denoted by G-OD(t,k,n).Consider a 2-idempotent 3-quasigroup as an OA(3,4,n), (N ,B), and discard

all ordered quadruples (x, y, z,w) with |{x, y, z,w}|≤2. Then the resulting arrayis an OD(3,4,n). Conversely, adding all ordered quadruples (x, x, x, x) (x ∈N )

and (x, x, y, y), (x, y, x, y), (x, y, y, x) (x, y∈N , x �= y) to the 4-N array of anOD(3,4,n) produces a 2-idempotent 3-quasigroup. Therefore, a 2-idempotent3-quasigroup of order n with a conjugate invariant subgroup G is equivalent to aG-OD(3,4,n). For simplicity, we will use the notation of OD consistently in whatfollows.One necessary condition for the existence of a G-OD(3,4,n) is that |G|·gcd(n−

3,12) must divide 24 (see [17]). Different names of G-OD(3,4,n) are used byresearchers corresponding to different subgroups of S4. For examples, an S4-OD(3,4,n)

is called totally symmetric and an S3-OD(3,4,n) is called commutative. Othersubgroups of S4 (i.e. I (identity group), A4 (alternating group), D4 (dihedral group),K4 (Klein 4-group), G2 (〈(13)(24)〉), C4 (〈(1234)〉), C3 (〈(123)(4)〉)) were consid-ered by many researchers. Known results of G-OD(3,4,n) are summarized asfollows.

Theorem 1.1.

(1) There exists an I -OD(3,4,n) for all n, n �=3,7 [17,18].(2) There exists an S4-OD(3,4,n) for n≡2,4(mod 6)) [4].(3) There exists an S3-OD(3,4,n) for n≡2, 4(mod 6) [1].(4) There exists an A4-OD(3,4,n) for n≡1,2,4,5,8,10(mod 12) [8].(5) There exists a D4-OD(3,4,n) for all even n or n=1 [17].(6) There exists a K4-OD(3,4,n) for n≡0,1,2(mod 4) [17].(7) There exists a G2-OD(3,4,n) n for all n,n �=3,7 [17].(8) There exists a C4-OD(3,4,n) for all n≡0,1,2(mod 4) with n �=5 [10,17].

Journal of Combinatorial Designs DOI 10.1002/jcd

294 JI AND WEI

The only unsolved case of determining the spectrum for G-OD(3,4,n) is when Gequals to C3. Since C3 is a subgroup of the alternating group A4, the known results ofA4-OD(3,4,n) imply some of C3-OD(3,4,n).

Theorem 1.2 (Teirlinck [17]). There exists a C3-OD(3,4,n) if n≡1,2,4,5,8,10(mod 12) with n≥4, and there is no C3-OD(3,4,7).

Since the necessary condition of the existence of a C3-OD(3,4,n) is n≡1,2(mod 3),the open cases are for orders n≡7,11(mod 12) with n≥11 [17]. The main purposeof this article is to construct a C3-OD(3,4,n) for all n≡7,11(mod 12) with n≥11.Together with the known results, this article will completely determine the spectrum forG-OD(3,4,n).The rest of the article is arranged as follows. Some definitions of other combinatorial

objects used in our constructions are given in Section 2. Sections 3 and 4 determinethe existence of a C3-OD(3,4,12k+7) and a C3-OD(3,4,12k+11), respectively. As acorollary given in Section 5, there is an overlarge set of Mendelsohn triple system oforder n if and only if n≡0,1(mod 3) and n �=6.

2. DEFINITIONS AND CONSTRUCTIONS

Candelabra quadruple systems (CQSs) are useful in the construction of Steiner quadruplesystems (for example see [7]). For our constructions, we make use of a similar config-uration, i.e. an ordered CQS, which has been introduced in [10].An ordered CQS of order v with a candelabra of type (ga11 . . .gakk :s) is a 4-X array

A with a branch set G and a stem S, where

1. X is a set of v=s+∑1≤i≤k ai gi points, S is an s-subset of X , and G={G1,G2, . . .}

is a partition of X \S (called groups) of type ga11 . . .gakk (there are a1 groups ofsize g1, . . . ,ak groups of size gk);

2. Each row contains 4 distinct elements of X such that any 3 columns of A containsevery ordered triple T =(a,b,c) of X with |{a,b,c}∩(S∪Gi )|<3 for all i exactlyonce and no ordered triples of S∪Gi for any i .

If the array A is invariant under the action of the subgroup of conjugations G, then it isdenoted by G-CQS(ga11 . . .gakk :s).Since C3 is a subgroup of the alternating group A4, the existence of an A4-CQS

implies the existence of a C3-CQS with the same type. In order for A4 to distinguish therepresentative of each orbit under the action of conjugation group C3, the representativeof the orbit generated by the 4-tuple (a,b,c,d) under the action of A4 is denoted by[a,b,c,d] in the following.

Lemma 2.1. There exists an A4-CQS(24 :3) and therefore a C3-CQS(24 :3).Proof. An A4-CQS(24 :3) is constructed on Z11 with the groups {i, i+4} (i ∈ Z4)

and a stem {8,9,10} as follows. The required 4-tuples are generated by the following4-tuples under the action of the group 〈(0 1 2 3 4 5 6 7)(8 9)(10)〉 and the



conjugation subgroup A4, where the 4-tuples with a star in the list will occur twice andwe shall take each only once.

[0 1 6 10] [0 7 2 10] [0 1 2 8] [1 4 7 8] [2 1 0 8] [7 4 1 8] [0 1 3 4][0 1 4 5]∗ [0 2 4 6]∗ �

Some other definitions are needed for our constructions. A holey ordered designOD(t,k,n) with a hole of size h is a k-N array A with a hole H , |N |=n, H an h-subsetof N satisfying the properties that each row of A contains k distinct elements, and everyt columns of A contain each ordered t-subset T of N , which is not an ordered t-subsetof H , exactly once and no ordered t-subsets of H . Further, if the array A is invariantunder the group of conjugations G, then it is briefly denoted by G-HOD(t,k,n;h).Using G-CQS, G-HOD and G-OD, we have the following.

Lemma 2.2 (Ji [10]). Suppose that there is a G-CQS(g10ga11 . . .garr :s). If there exist a

G-OD(g0+s) and a G-HOD(gi +s;s) for 1≤ i≤r, then there exists a G-OD(s+g0+∑1≤i≤r ai gi ).

An ordered transverse quadruple system of order v and type (ga11 ga22 . . .garr ) is a 4-Xarray Awith a group set G where |X |=∑

1≤i≤r ai gi , G={G1,G2, . . .} a partition of X oftype ga11 . . .garr , each of whose rows contains 4 elements from distinct groups, such thatin any three columns of A every ordered triple T =(a,b,c) of X with |{a,b,c}∩Gi |≤1for all i appears exactly once. If the array A is invariant under the action of the subgroupof conjugations G, then it is denoted by G-TRQS(ga11 . . .gakk ). Clearly, a G-OD(3,4,n)

is also a G-TRQS(1n).Let v and t be positive integers and K a set of positive integers. A group divisible

t-design (or t-GDD) of order v and block sizes from K denoted by GDD(t,K ,v) is atrio (X,G,B) where

1. X is a v-set of points,2. G is a partition of X into subsets (called groups),3. B is a family of subsets of X each of cardinality from K (called blocks) such that

each block intersects any given group in at most one point, and4. each t-set of points from t distinct groups is contained in exactly one block.

Similar to CQS, the group-type (or type) of a t-GDD is defined to be the multiset{|G| :G∈|G}.Clearly, an S4-TRQS is in 1-1 correspondence with a group divisible 3-design. Mills

[16] almost determined the existence of a GDD(3,4,mg) of type gm . Recently, mostof the last undecided infinite class m=5 and g≡2, 10(mod 12) was improved by thefirst author in [11].Lemma 2.3 (Ji [11] and [16]). For m≥4 and m �=5, a GDD(3,4,mg) of type gm

exits if and only if gm is even and g(m−1)(m−2) is divisible by 3. For m=5, aGDD(3,4,mg) of type gm exits if g is even, g �=2 and g �≡10,26(mod 48).

Since A4 is a subgroup of the symmetric group S4, the above lemma implies that thereis an A4-TRQSwith the same parameters. SinceC3 is a subgroup of the alternating groupA4, the existence of an A4-TRQS implies the existence ofC3-TRQS with the same type.


296 JI AND WEI

Since there is a GDD(3,4,4g) of type g4 containing a subdesign GDD(2,4,4g) forg≥4 and g �=6,10 by [8, Lemma 2.3], we have the following results from the proof of[8, Lemma 2.2].Lemma 2.4. For g≥4 and g �=6,10, there exist an A4-TRQS(g411) and an A4-CQS(g4 :1).Since there is a GDD(3,4,4g) of type g4 containing g pairwise disjoint subdesigns

GDD(2,4,4g) for g �≡2(mod 4) with g≥4 by [2, Theorem 4.1], we have the followingresults from the proof of [8, Lemma 2.2].Lemma 2.5. For g �≡2(mod 4) with g≥4, there exist an A4-CQS(g4 :s) and anA4-TRQS(g4s1) for any s≤g.

A t-wise balanced design (tBD) S(t,K ,n) is a GDD(t,K ,v) with group type 1v .When K ={k}, we simply write k for K . An S(3,4,n) is called a Steiner quadruplesystem (briefly by SQS(n)). An s-fan design is an (s+3)-tuple (X,G,B1,B2, . . . ,Bs,T ),where X is a finite set of points, G, Bi (1≤ i≤s) and T are all sets of subsets of Xwith the property that (X,G) is a 1BD, each (X,G∪Bi ) is a 2BD for 1≤ i≤s, and(X,G∪(

⋃si=1Bi )∪T ) is a 3BD. The members of G and (

⋃si=1Bi )∪T are called groups

and blocks, respectively. Let the type of (X,G) be ga11 ga22 . . .garr . If block sizes of Biand T are from Ki (1≤ i≤s) and KT , respectively, then the s-fan design is denoted bys-FG(3, (K1,K2, . . . ,Ks,KT ),

∑ri=1 ai gi ) of type ga11 ga22 . . .garr .

Lemma 2.6 (Ji [10]). Suppose that there exists an s-FG(3, (K1, . . . ,Ks,KT ),v)

of type ga11 . . .garr . If there exist a G-TRQS(bk) for any k∈KT , a G-TRQS(bki+1)

for any ki ∈Ki (2≤ i≤s) and a G-CQS(bk1 :e) for any k1∈K1, then there exists aG-CQS((bg1)a1 . . . (bgr )ar :e+sb−b).

For G= S4, the above lemma is in 1-1 correspondence with the usually recursiveconstruction for CQSs.

Lemma 2.7. If there exists a G-TRQS(gn), then there exists a G-TRQS((mg)n) forany positive integer m.

Proof. Let A be the set of 4-tuples of a given G-TRQS(gn) on N with a group set G.Let B consist of the following 4-tuples:

((x1, y1), (x2, y2), (x3, y3), (x4, y4)),

where (x1, x2, x3, x4)∈A, y1, y2, y3, y4∈ Zm , and y1+ y2+ y3+ y4≡0(mod m).It is routine to check that B is the set of 4-tuples of a G-TRQS((mg)4) on N×Zm

with the group set G′ ={H×Zm :H ∈G}. �

3. EXISTENCE OF C3-OD(3,4,12k+7)

In this and the next sections, we will construct the required designs. Both direct andrecursive constructions are used. In most direct constructions, we shall give base 4-tuplesand then use two kinds of actions to generate the rest of the array. One action is somepermutations based on N (for example, permutations on Zn). Then some conjugationsare used to form the other rows.



Lemma 3.1. There exists an A4-HOD(3,4,19;7).Proof. An A4-HOD(3,4,19;7) is constructed on Z19 with a hole {5,6,7,8,9,10,11}.Beside the 4-tuples of an A4-OD(3,4,5) on {0,1,2,3,4}, other 4-tuples are generated bythe following 4-tuples under the action of the group 〈(0)(1)(2)(3)(4)(5 6 7 8 9 10 11)(12 13 14 15 16 17 18)〉 and the conjugation subgroup A4.

[0 1 5 6] [0 1 12 13] [0 2 5 7] [0 2 12 14] [0 3 5 8] [0 3 12 15][0 4 5 12] [0 4 12 7] [0 5 9 12] [0 5 10 14] [0 5 11 13] [0 5 13 18][0 5 17 16] [0 5 18 15] [1 2 5 6] [1 2 12 13] [1 3 5 7] [1 3 12 14][1 4 5 14] [1 4 12 6] [1 5 8 12] [1 5 9 15] [1 5 10 17] [1 5 13 16][1 5 15 13] [1 5 17 14] [2 3 5 15] [2 3 12 8] [2 4 5 9] [2 4 12 15][2 5 6 18] [2 5 7 15] [2 5 8 17] [2 5 12 14] [2 5 16 13] [2 5 18 12][3 4 5 8] [3 4 12 18] [3 5 6 14] [3 5 7 16] [3 5 11 17] [3 5 12 13][3 5 15 18] [3 5 17 12] [4 5 6 13] [4 5 7 17] [4 5 10 18] [4 5 11 15][4 5 14 16] [4 12 15 14] [5 6 12 15] [5 6 13 17] [5 6 15 16] [5 6 17 12][5 7 14 13] [5 7 15 12] [5 7 17 18] [5 7 18 14] [5 8 12 13] [5 8 13 14][5 8 14 16] [5 8 16 18] [5 8 17 15] [5 14 15 17] [12 13 14 17] [12 13 15 18] �

Lemma 3.2. There exists a C3-OD(3,4,19).

Proof. AC3-OD(3,4,19) is constructed on Z19. The first part of the 4-tuples are gener-ated by the following 4-tuples under the action of the group 〈(0 1 2 3 4 5 6 7 8 910 11 12 13 14 15 16 17 18)〉 and the conjugation subgroup C3.

(1 7 11 0) (1 11 7 0) (2 3 14 0) (2 14 3 0) (4 6 9 0) (4 9 6 0)(18 12 8 0) (18 8 12 0) (17 16 5 0) (17 5 16 0) (15 13 10 0) (15 10 13 0)

The second part of the 4-tuples are generated by the following 4-tuples underthe action of the group 〈(0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18),(0)(1 7 11)(2 14 3) (4 9 6)(5 16 17)(8 18 12) (10 13 15)〉 and the conjugationsubgroup C3.

(0 1 2 5) (0 1 3 2) (0 1 4 13) (0 1 5 16) (0 1 6 11) (0 1 7 6)(0 1 9 14) (0 1 10 7) (0 1 11 9) (0 1 13 10) (0 1 14 8) (0 1 15 18)(0 1 16 15) (0 1 17 12) (0 2 1 11) (0 2 4 5) (0 2 6 4) (0 2 7 18)(0 2 10 9) (0 2 11 13) (0 2 14 17) (0 2 15 10) (0 4 1 13) (0 4 2 1)(0 4 3 10) (0 4 9 18) (0 4 14 6) (0 5 1 18) (0 5 4 6) (0 8 4 14) �

Lemma 3.3. There exist a C3-OD(3,4,60k+11) and a C3-OD(3,4,60k+31) for anypositive integer k.

Proof. Firstly, we construct an A4-CQS(103 :1) on (Z10×Z3)∪{∞} with the groupsZ10×{i}, i ∈ Z3 and a stem {∞}. The required 4-tuples are generated by the following4-tuples under the action of the conjugation subgroup A4, where i ∈ Z3, x, y, z∈ Z10,x+ y+z≡0(mod 10) and d∈{2,3,4}.[(x, i), (x+5, i), (y+i, i+1), (z, i+2)] [(x, i), (x+1, i), (y+7+i, i+1), (z, i+2)][(x, i), (x+1, i), (z+2, i+2), (y+i, i+1)] [(x,0), (y+6,1), (z,2),∞][(z+1,2), (y,1), (x,0),∞] [(x, i), (x+d, i), (y, i+1), (y+d, i+1)]


298 JI AND WEI

Secondly, start with a 1-FG(3, (3,4),6k+1) and a 1-FG(3, (3,4),6k+3), which canbe obtained by deleting one point from an SQS(6k+2) and an SQS(6k+4), respectively(see [7] for the existence of SQSs). Applying Lemma 2.6 with the above A4-CQS(103 :1)and known A4-TRQS(104) by Lemma 2.3, we obtain an A4-CQS(106k+1 :1) and anA4-CQS(106k+3 :1).Finally, applying Lemma 2.2 with the known C3-OD(3,4,11) by Lemma 4.1, we get

the result. �

Lemma 3.4. There exists an A4-CQS(63 :1).Proof. An A4-CQS(63 :1) is constructed on (Z6×Z3)∪{∞} with the groups Z6×{i},i ∈ Z3 and a stem {∞}. The required 4-tuples are generated by the following 4-tuplesunder the action of the conjugation subgroup A4, where i ∈ Z3, x, y, z∈ Z6, x+ y+z≡0(mod 6).

[(x, i), (x+3, i), (y+2i, i+1), (z, i+2)] [(x,0), (y+3,1), (z,2),∞][(z,2), (y,1), (x,0),∞] [(x, i), (x+2, i), (y, i+1), (y+2, i+1)][(x+1, i), (x+2, i), (y, i+1), (z, i+2)] where y≡ z (mod 3)[(x+4, i), (x+5, i), (z, i+2), (y, i+1)] where y≡ z (mod 3)[(x, i), (x+1, i), (y, i+1), (y+1, i+1)] where x≡ y (mod 3)[(x, i), (x+1, i), (y+1, i+1), (y, i+1)] where x≡ y (mod 3) �

Lemma 3.5. There exists a C3-OD(3,4,12k+7) for any positive integer k.

Proof. Start with a 2-FG(3, (3,3,{4,6}),2k) of type 2k−241 if k≡2(mod 3), or of type2k if k≡0,1(mod 3), which exists in the proof of [8, Theorem 2.7]. Applying Lemma2.6 with the known A4-CQS(63 :1) by Lemma 3.4 and A4-TRQS(64) by Lemma 2.3,we obtain an A4-CQS((12)k−2241 :7) or an A4-CQS(12k :7). The conclusion followsfrom Lemma 2.2 with the known A4-HOD(3,4,19;7) by Lemma 3.1, C3-OD(3,4,19)by Lemma 3.2 and C3-OD(3,4,31) by Lemma 3.3. �

4. EXISTENCE OF C3-OD(3,4,12k+11)

Lemma 4.1. There exist a C3-HOD(3,4,11;5) and a C3-OD(3,4,11).

Proof. A C3-HOD(3,4,11;5) is constructed on Z11 with a hole {0,1,2,3,4}.The conjugation subgroup C3 acts on the following 4-tuples to generate the first part

of the 4-tuples:

(5 6 7 0) (5 7 6 1) (8 9 10 0) (8 10 9 1).

The other 4-tuples are generated by the following 4-tuples under the action of thegroup 〈(0)(1)(2 3 4)(5 6 7)(8 9 10)〉 and the conjugation subgroup C3.

(0 1 5 6) (0 1 8 9) (0 2 5 7) (0 2 6 8) (0 2 7 10) (0 2 8 6)(0 2 9 5) (0 2 10 9) (0 5 1 7) (0 5 2 6) (0 5 3 10) (0 5 4 9)(0 5 6 1) (0 5 7 8) (0 5 8 2) (0 5 9 4) (0 5 10 3) (0 8 1 10)(0 8 2 5) (0 8 3 9) (0 8 4 6) (0 8 5 2) (0 8 6 4) (0 8 7 3)



(0 8 9 1) (0 8 10 7) (1 2 5 8) (1 2 6 10) (1 2 7 9) (1 2 8 7)(1 2 9 6) (1 2 10 5) (1 5 2 10) (1 5 3 9) (1 5 4 8) (1 5 6 7)(1 5 7 3) (1 5 8 0) (1 5 9 2) (1 5 10 4) (1 8 2 6) (1 8 3 5)(1 8 4 7) (1 8 5 3) (1 8 6 0) (1 8 7 4) (1 8 9 10) (1 8 10 2)(2 3 5 9) (2 3 6 8) (2 3 7 6) (2 3 8 7) (2 3 9 10) (2 3 10 5)(2 4 5 8) (2 4 6 7) (2 4 7 5) (2 4 8 10) (2 4 9 6) (2 4 10 9)(2 5 6 3) (2 5 7 4) (2 5 8 9) (2 5 9 0) (2 5 10 1) (2 6 5 7)(2 6 7 3) (2 6 8 0) (2 6 9 1) (2 6 10 4) (2 7 5 3) (2 7 6 0)(2 7 8 1) (2 7 9 4) (2 7 10 6) (2 8 5 4) (2 8 6 5) (2 8 7 1)(2 8 9 3) (2 8 10 0) (2 9 5 0) (2 9 6 1) (2 9 7 10) (2 9 8 4)(2 9 10 3) (2 10 5 1) (2 10 6 4) (2 10 7 0) (2 10 8 3) (2 10 9 8)(5 6 8 10) (5 6 9 8) (5 6 10 9) (5 7 8 9) (5 7 9 10) (5 7 10 2)(5 8 9 6) (5 8 10 4) (5 9 8 7) (5 9 10 6) (5 10 8 6) (5 10 9 7)

Filling the hole with an A4-OD(3,4,5) by Theorem 1.1, we get a C3-OD(3,4,11). �

Lemma 4.2. There exist a C3-OD(3,4,24k+11) and a C3-HOD(3,4,24k+11;5) forany non-negative integer k.

Proof. For k=0, the result holds by Lemma 4.1.For k>0, deleting one point from a 1-FG(3, (4,4),12k+4) of type 112k+4 [9] yields

a 2-FG(3, (4,3,4),12k+3) of type 34k+1. Applying Lemma 2.6 with the knownA4-CQS(24 :3) by Lemma 2.1 and an A4-TRQS(24) by Lemma 2.3, we obtain anA4-CQS(64k+1 :5), which is also a C3-CQS(64k+1 :5). Since there are a C3-HOD(3,4,11;5) and a C3-OD(3,4,11) by Lemma 4.1, there are a C3-OD(3,4,24k+11) and a C3-HOD(3,4,24k+11;5) by Lemma 2.2. �

Lemma 4.3. There exists a C3-OD(3,4,72k+23) for any non-negative integer k.

Proof. For k=0, a C3-OD(3,4,23) is constructed on Z23. Its 4-tuples are generated bythe following 4-tuples under the action of the group 〈(0 1 2 3 4 5 6 7 8 9 10 11 1213 14 15 16 17 18 19 20), (0)(1 2 4 8 16 9 18 13 3 6 12)(5 10 20 17 11 2221 19 15 7 14)〉 and the conjugation subgroup C3.

(0 1 2 3) (0 1 3 4) (0 1 4 5) (0 1 5 11) (0 1 6 18) (0 1 7 12) (0 1 8 6)(0 1 9 16) (0 1 12 17) (0 1 13 20) (0 1 16 10) (0 1 18 7) (0 5 1 8) (0 5 2 21)

For k>0, we first construct an A4-CQS((24k+6)3 :5) on (Z24k+6×Z3)∪({∞}×Z5)

with the groups Z24k+6×{i}, i ∈ Z3 and a stem {∞}×Z5. The required 4-tuplesare generated by the following 4-tuples under the action of the conjugationsubgroup A4, where i ∈ Z3, x, y, z∈ Z24k+6, x+ y+z≡0(mod 24k+6), 2≤a≤4k and d∈{2,3,4, . . . ,12k}\({6 j−2 :2≤ j ≤2k}∪{24k+8−6 j :2k+1≤ j≤4k})={3,5, . . . ,12k−1,6,12,18, . . . ,12k}.

[(x, i), (x+6a−2, i), (y+i+12k−3a+6, i+1), (z, i+2)][(x, i), (x+6a−2, i), (z, i+2), (y+i+12k−3a+6, i+1)][(x, i), (x+12k+3, i), (y+i+3, i+1), (z, i+2)] [(x,0), (y+1,1), (z,2), (∞,0)][(x,0), (y+3,1), (∞,0), (z,2)] [(x,0), (y+12k+3,1), (z,2), (∞,1)]


300 JI AND WEI

[(x,0), (y+12k+3,1), (∞,1), (z,2)] [(x,0), (y+12k+9,1), (z,2), (∞,2)][(x,0), (y+12k+9,1), (∞,2), (z,2)] [(x,0), (y+12k+6,1), (z,2), (∞,3)][(x,0), (y+5,1), (∞,3), (z,2)] [(x,0), (y+12k+8,1), (z,2), (∞,4)][(x,0), (y+12k+4,1), (∞,4), (z,2)] [(x, i), (x+d, i), (y, i+1), (y+d, i+1)][(x+2, i), (x+1, i), (y, i+1), (z, i+2)] where y≡ z (mod 3)

[(x+12k+5, i), (x+12k+4, i), (z, i+2), (y, i+1)] where y≡ z (mod 3)

[(x, i), (x+1, i), (y, i+1), (y+1, i+1)] where x≡ y (mod 3)

[(x, i), (x+1, i), (y+1, i+1), (y, i+1)] where x≡ y (mod 3)

[(x+12k+7, i), (x+2, i), (y, i+1), (z, i+2)] where y≡ z (mod 3)

[(x+4, i), (x+12k+5, i), (z, i+2), (y, i+1)] where y≡ z (mod 3)

[(x, i), (x+12k+1, i), (y, i+1), (y+12k+1, i+1)] where x≡ y (mod 3)

[(x, i), (x+12k+1, i), (y+12k+1, i+1), (y, i+1)] where x≡ y (mod 3)

Since there are a C3-OD(3,4,24k+11) and a C3-HOD(3,4,24k+11;5) by Lemma 4.2,there exists a C3-OD(3,4,72k+23) by Lemma 2.2. �

Lemma 4.4. There exist a C3-OD(3,4,47) and a C3-HOD(3,4,47;11).Proof. Firstly, we construct an A4-TRQS(3421) on Z12∪{x, y} with the groups {i, i+4, i+8}, i ∈ Z4 and {x, y}. The required 4-tuples are generated by the following 4-tuplesunder the action of the group 〈(0 1 2 3 4 5 6 7 8 9 10 11)(x y)〉 and the conjugationsubgroup A4, where the 4-tuples with a star in the list will occur twice and we shalltake each only once.

[0 2 3 5] [0 2 5 3] [0 1 2 x] [0 1 x 2] [1 6 11 x] [1 6 x 11][0 3 6 x] [1 10 7 x] [0 1 6 7]∗ [0 1 7 6]∗

For each pair of groups, construct an A4-CQS(32 :0), which exists from the proof of[8, Theorem 2.1]. An A4-CQS(34 :2) is then obtained.Secondly, start with a 1-FG(3, (4,{4,5}),15) of type 35, which can be obtained by

deleting two points from an inversive plane of order 4. Applying Lemma 2.6 with theabove A4-CQS(34 :2) and A4-TRQS(3l) for l=4,5, which can be obtained by applyingLemma 2.7 with the known A4-TRQS(1l) by Theorem 1.1, we obtain an A4-CQS(95:2).Finally, we obtain a C3-OD(3,4,47) and a C3-HOD(3,4,47;11) by Lemma 2.2 with

the known C3-OD(3,4,11) by Lemma 4.1. �


Proof. For k=0, the result holds by Lemma 4.4.For k>0, we first construct an A4-CQS(363 :11) and an A4-CQS(365 :11).An A4-CQS(363 :11) is constructed on (Z36×Z3)∪{(∞,0), (∞,1), . . . , (∞,10)}

with the groups Z36×{i}, i ∈ Z3 and a stem {(∞,0), (∞,1), . . . , (∞,10)}. Its 4-tuplesare generated by the following 4-tuples under the action of the conjugation subgroup A4,



where i ∈ Z3, x, y, z∈ Z36, x+ y+z≡0(mod 36),a∈{2,3,4} and d∈{2,3,4, . . . ,15}\{10,14}.

[(x, i), (x+6a−2, i), (y+i+21−3a, i+1), (z, i+2)][(x, i), (x+6a−2, i), (z, i+2), (y+i+21−3a, i+1)][(x, i), (x+18, i), (y+i, i+1), (z, i+2)] [(x,0), (y+3+i,1), (z,2), (∞, i)][(x,0), (y+3+i,1), (∞, i), (z,2)] [(x,0), (y+6+i,1), (z,2), (∞,3+i)][(x,0), (y+6+i,1), (∞,3+i), (z,2)] [(x,0), (y+21+i,1), (z,2), (∞,6+i)][(x,0), (y+21+i,1), (∞,6+i), (z,2)] [(x,0), (y+24,1), (z,2), (∞,9)][(x,0), (y+24,1), (∞,9), (z,2), ] [(x,0), (y,1), (∞,10), (z,2)][(x,0), (y,1), (z+18,2), (∞,10)] [(x, i), (x+d, i), (y, i+1), (y+d, i+1)][(x+34, i), (x+35, i), (y, i+1), (z, i+2)] where y≡ z (mod 3)

[(x+34, i), (x+35, i), (z, i+2), (y, i+1)] where y≡ z (mod 3)



[(x+2, i), (x+19, i), (z, i+2), (y, i+1)] where y≡ z (mod 3)

[(x+20, i), (x+1, i), (y, i+1), (z, i+2)] where y≡ z (mod 3)



Start with a 1-FG(3, (4,{4,5}),15) of type 35 and apply Lemma 2.6 with the knownA4-CQS(124 :11) by Lemma 2.5 and A4-TRQS(36l) for l=4,5 by Lemma 2.3. AnA4-CQS(365 :11) is then obtained.Secondly, start with a 1-FG(3, ({3,5},{4,6}),2k+1) of type 12k+1, which can be

obtained by deleting one point from an S(3,{4,6},2k+2) [5]. Apply Lemma 2.6 withthe above A4-CQS(36a :11) and A4-TRQS(12a+1) for a∈{3,5} by Lemma 2.3. AnA4-CQS(362k+1 :11) is then obtained.Finally, we apply Lemma 2.2 with the known A4-OD(3,4,47) and A4-HOD(3,4,

47;11) by Lemma 4.4 to obtain the result. �

Now, we need some designs of SQS(v) with a subdesign SQS(22).

Lemma 4.6. There exists an SQS(36k+8) with a subdesign SQS(22) for any positiveinteger k with k �=2.

Proof. For k=1, applying the doubling construction for SQS with the known SQS(22)we get the result. For each k≥3, since there are an SQS(44)with a subdesign SQS(8) andan SQS(80) with a subdesign SQS(8) [14], there is an SQS(36k+8) with a subdesignSQS(44) by [6, Theorem 5.7]. The result is then obtained by replacing the subdesignby an SQS(44) containing a subdesign SQS(22). �


302 JI AND WEI

Lemma 4.7. There exists an SQS(36k+26)with a subdesign SQS(22) for any positiveinteger k with k �=2.

Proof. For k=1, there is a CQS(203 :2) [14]. Construct an SQS(22) on the unions ofeach group and the stem. An SQS(36k+26) with a subdesign SQS(22) is then obtained.For k≥3, deleting one point from an S(3,{4,6},2k+2) [5] gives a 1-FG(3, ({3,5},

{4,6}),2k+1) of type 12k−231. Apply Lemma 2.6 with the known GDD(3,4,18b) oftype 18b for b=4,6 by Lemma 2.3, CQS(183 :8) [14] and CQS(185 :8), which can beobtained by applying Lemma 2.6 with the known 1-FG(3, (4,{4,5}),15) of type 35,CQS(64 :8) [3] and GDD(3,4,6b) of type 6b for b=4,5 by Lemma 2.3. Then we obtaina CQS(182k−2541 :8). Construct an SQS(26) with a subdesign SQS(8) on the unionsof each group of size 18 and the stem such that the subdesign is based on the stem.We then obtain an SQS(36k+26) with a subdesign SQS(62). Replacing the subdesignSQS(62) by an SQS(62) with a subdesign SQS(22) we obtain an SQS(36k+26) witha subdesign SQS(22). �


Proof. For k=0, it exists by Lemma 3.3.For k=2, start with a 1-FG(3, ({4,5},{4,5}),17) of type 117 by [12, Lemma 3.8].

By Lemmas 2.3 and 2.5 there is an A4-TRQS(21a) for a=4,5. By Lemma 2.5 thereis an A4-CQS(214 :2). An A4-CQS(215 :2) can be obtained by applying Lemma 2.6with the known 1-FG(3, (4,{4,5}),15) of type 35, A4-CQS(74 :2) by Lemma 2.5 andA4-TRQS(7b) of type 7b for b=4,5 by Lemmas 2.3 and 2.5. Use these designs asinput designs and apply Lemma 2.6. Then we obtain an A4-CQS(2117 :2). ApplyingLemma 2.2 with the known C3-OD(3,4,23) we obtain a C3-OD(3,4,144k+71).For k>0 and k �=2, by Lemma 2.5 there is an A4-CQS((36k+15)4 :11). By

Lemma 4.7, there is an SQS(36k+26) with a subdesign SQS(22), which is in 1-1correspondence with an S4-OD(3,4,36k+26) containing a subdesign S4-OD(3,4,22).Replacing the subdesign S4-OD(3,4,22) with a C3-HOD(3,4,22;11) gives a C3-OD(3,4,36k+26;11), where a C3-HOD(3,4,22;11) can be obtained by constructing aC3-OD(3,4,11) on a group of an A4-CQS(112 :0) [13, Lemma 2.2]. Then we obtainthe result by Lemma 2.2. �


Proof. For k=1, start with a 1-FG(3, (4,{4,5}),15) of type 35. Apply Lemma 2.6with the known A4-CQS(194 :2) by Lemma 2.5, an A4-TRQS(194) by Lemma 2.3and an A4-TRQS(195) by Lemma 2.5. Then we obtain an A4-CQS(575 :2). ApplyingLemma 2.2 with the known C3-OD(3,4,59) by Lemma 4.2 we get the result.For k �=1, by Lemma 2.5 there is an A4-CQS((36k+33)4 :11). By Lemma 4.6, there

is an SQS(36k+44) with a subdesign SQS(22), which is in 1-1 correspondence with anS4-OD(3,4,36k+8) containing a subdesign S4-OD(3,4,22). Replacing the subdesignwith the known C3-HOD(3,4,22;11), we get a C3-HOD(3,4,36k+44;11). Then weobtain the result by using Lemma 2.2. �

Combining Lemmas 4.2, 4.3, 4.5, 4.8 and 4.9, we obtain our main result of thissection.




5. CONCLUSION

Combining Theorem 1.2, Lemmas 3.5 and 4.10, we obtain our main result of this article.

Theorem 5.1. There is a C3-OD(3,4,n) if and only if n≡1,2(mod 3), n≥4 andn �=7.

Together with Theorem 1.1, the spectrum of 2-idempotent 3-quasigroups with conju-gate invariant subgroups is completely determined.As a corollary of Theorem 5.1, we can obtain the following result about overlarge

set of Mendelsohn triple systems. An OD(2,3,n) whose conjugate invariant subgroupcontains (1 2 3) is easily seen to be equivalent to a decomposition of the completesymmetric directed graph on a set of n vertices into cyclic triples. Such a decompositionis called a Mendelsohn triple system of order n and denoted by MTS(n). The MTS(n)

is known to exist for n≡0,1(mod 3) and n �=6 [15].An overlarge set of MTS(n), denoted by OLMTS(n), is a collection {(X \{x},Bx ) :

x ∈ X}, where X is an (n+1)-set, each Bx is a set of cyclic triples of X \{x} suchthat each (X \{x},Bx ) is an MTS(n) and

⋃x∈X Bx contains each cyclic triple of X

precisely once. This overlarge set is known to exist for n≡1,2,4,5,8,10(mod 12) withn≥4 [17].Suppose a C3-OD(3,4,n+1) (X,B) is given. Let Bx ={(a,b,c) :(a,b,c, x)∈B}

for any x ∈ X . Clearly, each (X \{x},Bx ) is an OD(2,3,n) whose conjugate invariantsubgroup contains (1 2 3). The collection {(X \{x},Bx ) : x ∈ X} is an OLMTS(n).Together with the necessary condition for the existence of an MTS, the spectrum for

C3-OD(3,4,n) leads to the spectrum for OLMTS.

Corollary 5.2. There exists an OLMTS(n) if and only if for n≡0,1(mod 3) andn �=6.

ACKNOWLEDGMENT

We thank the referees for their helpful comments. The first author would also like to thankthe hospitality of Lakehead University where the research was done during his visit.

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the spectrum of 2-idempotent 3-quasigroups with conjugate invariant subgroups

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