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ISRN Algebra
VolumeΒ 2012Β (2012), Article IDΒ 367129, 19 pages
http://dx.doi.org/10.5402/2012/367129
Research Article

On the Existence of (𝑣,π‘˜,πœ†) Difference Sets with π‘˜<1250 and π‘˜βˆ’πœ† Is a Square

Division of Sciences and Mathematics, Department of Mathematics, Livingstone College, Salisbury, NC 28144, USA

Received 20 January 2012; Accepted 8 February 2012

Academic Editors: A. V.Β Kelarev, D.Β Kressner, H.Β You, and A.Β Zimmermann

Copyright Β© 2012 Adegoke Solomon Osifodunrin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We combine Turyn's self-conjugacy result, variance technique, Dillon dihedral trick, and Sylow theorem to investigate the existence of (𝑣,π‘˜,πœ†) difference sets in which π‘˜βˆ’πœ† is a square and π‘˜<1250.

1. Introduction

Let 𝐺 be a multiplicative group of order 𝑣 and let 𝐷 be a subset of 𝐺 consisting of π‘˜ elements, where 1<π‘˜<π‘£βˆ’1. 𝐷 is a nontrivial (𝑣,π‘˜,πœ†) difference set if every nonidentity element can be reproduced πœ† times by the multiset {𝑑1𝑑2βˆ’1βˆΆπ‘‘1,𝑑2∈𝐷,𝑑1≠𝑑2}. The natural number 𝑛=π‘˜βˆ’πœ†>1 is known as the order of the difference set. The group structure determines the nature of the difference set. For instance, if the underlying group 𝐺 is abelian (resp., nonabelian or cyclic), then 𝐷 is abelian (resp., nonabelian or cyclic) difference set. The study of difference sets integrates various techniques ranging from algebraic number theory to geometry, algebra, and combinatorics [1]. There are many classical results on constructions and nonexistence of difference sets in the literature [2–14]. These results are mainly based on Hall’s multiplier concept [15] or Turyn’s self-conjugacy method [14]. Recently, Schmidt [13] developed a new method for studying combinatorial structures using group ring equations without any restrictive assumptions. Arasu [2], Arasu and Sehgal [3, 16], Baumert [4], Hughes [17], Iiams [7], Kibler [18], Kopilovich [10], Lander [11], and LΓ³pez and SΓ‘nchez [19] among other authors studied the existence of abelian (𝑣,π‘˜,πœ†) difference sets with π‘˜β‰€150. They were able to either indicate the existence or otherwise of difference sets. Some of these authors also listed parameter sets that were open, whose existence or otherwise has been concluded by other authors. This paper mainly uses Turyn’s self-conjugacy approach to study a class of (𝑣,π‘˜,πœ†) difference sets in which 𝑛=π‘š2, where π‘š is a positive integer. We illustrate with examples where 2β‰€π‘šβ‰€25 and the ideal generated by prime divisors of π‘š factors trivially in the respective cyclotomic rings. This assumption along with Dillon dihedral trick and Sylow theorems provide sufficient information required to decide the nonesxistence of the difference sets in some or all groups of order 𝑣.

We assume that 𝐺 is a finite group of order 𝑣. Section 2 gives a brief description of some basic results which include materials from group theory, representation, and algebraic number theories. Section 3 lists difference sets parameters that do not exist and examples of partial results of nonexistence of difference sets in groups of order 𝑣.

2. Preliminaries

2.1. Difference Sets

Let β„€ be the ring of integers and β„‚ be the field of complex numbers. Suppose that 𝐺 is a group of order 𝑣 and 𝐷 is a (𝑣,π‘˜,πœ†) difference set in 𝐺. We sometimes view the elements of 𝐷 as members of the group ring β„€[𝐺], which is a subring of the group algebra β„‚[𝐺]. Thus, 𝐷 represents both subset of 𝐺 and element βˆ‘π‘”βˆˆπ·π‘” of β„€[𝐺]. The sum of inverses of elements of 𝐷 is 𝐷(βˆ’1)=βˆ‘π‘”βˆˆπ·π‘”βˆ’1. Consequently, 𝐷 is a difference set if and only if 𝐷𝐷(βˆ’1)=𝑛+πœ†πΊ,𝐷𝐺=π‘˜πΊ.(2.1) Suppose that 𝐷 is a difference set in a group 𝐺 of order 𝑣 and 𝑁 is a normal subgroup of 𝐺. Suppose that πœ“βˆΆπΊβ†’πΊ/𝑁 is a homomorphism. We can extend πœ“ by linearity, to the corresponding group rings. Thus, the difference set image in 𝐺/𝑁 (also known as the contraction of 𝐷 with respect to the kernel 𝑁) is the multiset 𝐷/𝑁=πœ“(𝐷)={π‘‘π‘βˆΆπ‘‘βˆˆπ·}. Let π‘‡βˆ—={1,𝑑1,…,π‘‘β„Ž} be a left transversal of 𝑁 in 𝐺. We can write βˆ‘πœ“(𝐷)=π‘‘π‘—βˆˆπ‘‡βˆ—π‘‘π‘—π‘‘π‘—π‘, where the integer 𝑑𝑗=|π·βˆ©π‘‘π‘—π‘| is known as the intersection number of 𝐷 with respect to 𝑁. In this work, we will always use the notation 𝐷 for πœ“(𝐷) and π‘šπ‘–β‰₯0 denotes the number of times 𝑑𝑖 equals 𝑖.

2.2. Representation and Algebraic Number Theories

A β„‚-representation of 𝐺 is a homomorphism, πœ’βˆΆπΊβ†’GL(𝑑,β„‚), where GL(𝑑,β„‚) is the group of invertible 𝑑×𝑑 matrices over β„‚. The positive integer 𝑑 is the degree of πœ’. A linear representation (character) is a representation of degree one. The set of all linear representations of 𝐺 is denoted by πΊβˆ—. β€‰πΊβˆ— is an abelian group under multiplication and if 𝐺′ is the derived group of 𝐺, then πΊβˆ— is isomorphic to 𝐺/𝐺′. Define πœπ‘šξ…žβˆΆ=𝑒(2πœ‹/π‘šβ€²)𝑖 to be a primitive π‘šξ…žth root of unity and πΎπ‘šξ…žβˆΆ=β„š(πœπ‘šξ…ž) to be the cyclotomic extension of the field of rational numbers, β„š, where π‘šβ€² is the exponent of 𝐺. Without loss of generality, we may replace β„‚ by the field πΎπ‘šξ…ž. Thus, the central primitive idempotents in β„‚[𝐺] is π‘’πœ’π‘–=πœ’π‘–(1)||𝐺||ξ“π‘”βˆˆπΊπœ’π‘–(𝑔)π‘”βˆ’1=1||𝐺||ξ“π‘”βˆˆπΊπœ’π‘–(𝑔)𝑔,(2.2) where πœ’π‘– is an irreducible character of 𝐺.

Aliases are members of group ring which enable us to transfer information from β„‚[𝐺] to group algebra β„š[𝐺] and then to β„€[𝐺]. Let 𝐺 be an abelian group and let Ξ©={πœ’1,πœ’2,…,πœ’β„Ž}, be the set of characters of 𝐺. The element π›½βˆˆβ„€[𝐺] is known as Ξ©-alias if for π΄βˆˆβ„€[𝐺] and all πœ’π‘–βˆˆΞ©, πœ’π‘–(𝐴)=πœ’π‘–(𝛽). Since βˆ‘π΄=πœ’βˆˆπΊβˆ—πœ’(𝐴)π‘’πœ’, we can replace the occurrence of πœ’(𝐴), which is a complex number by Ξ©-alias, 𝛽, an element of β„€[𝐺]. Furthermore, two characters of 𝐺 are algebraic conjugate if and only if they have the same kernel and we denote the set of equivalence classes of πΊβˆ— by πΊβˆ—/∼. The central rational idempotents in β„š[𝐺] are obtained by summing over the equivalence classes 𝑋𝑖={π‘’πœ’π‘–βˆ£πœ’π‘–βˆΌπœ’π‘—}βˆˆπΊβˆ—/∼ on the π‘’πœ’'s under the action of the Galois group of πΎπ‘šξ…ž over β„š. That is, [π‘’πœ’π‘–βˆ‘]=π‘’πœ’π‘—βˆˆπ‘‹π‘–π‘’πœ’π‘—, 𝑖=1,…,𝑠.

For instance, suppose 𝐺=πΆπ‘π‘šβ€²=⟨π‘₯∢π‘₯π‘π‘šβ€²=1⟩ (𝑝 is prime) is a cyclic group whose characters are of the form πœ’π‘–(π‘₯)=πœπ‘–π‘π‘šβ€², 𝑖=0,…,π‘π‘šξ…žβˆ’1. Then the rational idempotents are ξ€Ίπ‘’πœ’0ξ€»=1π‘π‘šβ€²ξ‚ƒπ‘’βŸ¨π‘₯⟩,πœ’π‘π‘—ξ‚„=1𝑝𝑗+1𝑝π‘₯π‘π‘šβ€²βˆ’π‘—ξ‚­βˆ’ξ‚¬π‘₯π‘π‘šβ€²βˆ’π‘—βˆ’1,0β‰€π‘—β‰€π‘šβ€²βˆ’1.(2.3) The following is the general formula employed in the search of difference set [22].

Theorem 2.1. Let 𝐺 be an abelian group and πΊβˆ—/∼ be the set of equivalence classes of characters. Suppose that {πœ’0,πœ’1,…,πœ’π‘ } is a system of distinct representatives for the equivalence classes of πΊβˆ—/∼. Then for π΄βˆˆβ„€[𝐺], one has 𝐴=𝑠𝑖=0π›Όπ‘–ξ€Ίπ‘’πœ’π‘–ξ€»,(2.4) where 𝛼𝑖 is any πœ’π‘–-alias for 𝐴.

Equation (2.4) is known as the rational idempotent decomposition of 𝐴.

Suppose that πœ’ is any nontrivial representation of degree 𝑑 and ξπœ’(𝐷)βˆˆβ„€[𝜁], where 𝜁 is the primitive root of unity. Suppose that π‘₯∈𝐺 is a nonidentity element. Then, πœ’(π‘₯𝐺)=πœ’(π‘₯)πœ’(𝐺)=πœ’(𝐺). This shows that (πœ’(π‘₯)βˆ’1)πœ’(𝐺)=0. Since π‘₯ is not an identity element, (πœ’(π‘₯)βˆ’1)β‰ 0 and πœ’(𝐺)=0 (β„€[𝜁] is an integral domain). Consequently, πœ’(𝐷)πœ’(𝐷)=𝑛⋅𝐼𝑑+πœ†πœ’(𝐺)=𝑛⋅𝐼𝑑, where 𝐼𝑑 is the 𝑑×𝑑 identity matrix. Furthermore, if πœ’ is a nontrivial representation of 𝐺/𝑁 of degree 𝑑 then 𝐷𝐷(βˆ’1)=𝑛⋅1𝐺/𝑁+|𝑁|πœ†(𝐺/𝑁) and ξπœ’(𝐷)ξπœ’(𝐷)=𝑛⋅𝐼𝑑.

Recall that the ring of integers of the cyclotomic field β„š[πœπ‘šξ…ž] is β„€[πœπ‘šξ…ž]. This ring is also an integral domain. Let 𝑝, π‘Ž, π‘βˆˆβ„€[πœπ‘šξ…ž]. The number 𝑝 is irreducible if 𝑝=π‘Žπ‘ implies one of π‘Ž or 𝑏 is a unit. The element 𝑝 is prime if π‘βˆ£π‘Žπ‘ implies π‘βˆ£π‘Ž or π‘βˆ£π‘ [23]. A domain is a unique factorization domain (UFD) if factorization into irreducibles is possible and unique. In UFD, the irreducibles are also primes. In order to successfully obtain the difference set images, we need the aliases. Suppose that 𝐺/𝑁 is an abelian factor group of exponent π‘šβ€² and 𝐷 is a difference set image in 𝐺/𝑁. If πœ’ is not a principal character of 𝐺/𝑁, then ξπœ’(𝐷)ξπœ’(𝐷)=𝑛 is an algebraic equation in β„€[πœπ‘šξ…ž]. The determination of the alias requires the knowledge of how the ideal generated by ξπœ’(𝐷) factors in cyclotomic ring β„€[πœπ‘šξ…ž], where πœπ‘šξ…ž is the π‘šβ€²th root of unity. If ξπ›ΏβˆΆ=πœ’(𝐷), then by (2.4), we seek π›Όβˆˆβ„€[𝐺/𝑁] such that πœ’(𝛼)=𝛿. The task of solving the algebraic equation 𝛿𝛿=𝑛 is sometimes made easier if we consider the factorization of principal ideals βŸ¨π›ΏβŸ©βŸ¨π›ΏβŸ©=βŸ¨π‘›βŸ©. Suppose we are able to find βˆ‘π›Ώ=πœ™(π‘šξ…ž)βˆ’1𝑖=0π‘‘π‘–πœπ‘–π‘šξ…žβˆˆβ„€[πœπ‘šξ…ž] such that 𝛿𝛿=𝑛, where πœ™ is the Euler πœ™-function. A theorem due to Kronecker [12, 13] states that any algebraic integer whose all conjugates have absolute value 1 must be a root of unity. We use this theorem to characterize the solutions. If there is any other solution to the algebraic equation, then it must be of the form 𝛿′=𝛿𝑒, where 𝑒=Β±πœπ‘—π‘šξ…ž is a unit.

The following result is used to determine the number of factors of an ideal in a ring: suppose 𝑝 is any prime and π‘šβ€² is an integer such that gcd(𝑝,π‘šβ€²)=1. Suppose that 𝑑 is the order of 𝑝 in the multiplicative group β„€βˆ—π‘šξ…ž of the modular number ring β„€π‘šξ…ž. Then the number of prime ideal factors of the principal ideal βŸ¨π‘βŸ© in the cyclotomic integer ring β„€[πœπ‘šξ…ž] is πœ™(π‘šβ€²)/𝑑, where πœ™ is the Euler πœ™-function, that is, πœ™(π‘šβ€²)=|β„€βˆ—π‘šξ…ž| [25]. For instance, the ideal generated by 2 has two factors in β„€[𝜁7], the ideal generated by 7 has two factors in β„€[𝜁20], while the ideal generated by 3 has four factors in β„€[𝜁40]. On the other hand, since 2𝑠 is a power of 2, the ideal generated by 2 is said to completely ramifies as power of ⟨1βˆ’πœ2π‘ βŸ©=⟨1βˆ’πœ2π‘ βŸ© in β„€[𝜁2𝑠].

According to Turyn [14], an integer 𝑛 is said to be semi-primitive modulo π‘šβ€² if for every prime factor 𝑝 of 𝑛, there is an integer 𝑖 such that π‘π‘–β‰‘βˆ’1modπ‘šβ€². In this case, βˆ’1 belongs to the multiplicative group generated by 𝑝. Furthermore, 𝑛 is self-cosnjugate modulo π‘šβ€² if every prime divisor of 𝑛 is semi primitive modulo π‘šξ…žπ‘, where π‘šξ…žπ‘ is the largest divisor of π‘šβ€² relatively prime to 𝑝. This means that all prime ideals over 𝑛 in β„€[πœπ‘šξ…ž] are fixed by complex conjugation. For instance, 72β‰‘βˆ’1(modπ‘šβ€²), where π‘šβ€²=2, 5, 10 and 7β‰‘βˆ’1(modπ‘šβ€²), π‘šβ€²=2, 4, 8. Thus, ⟨7⟩ is fixed by conjugation in β„€[πœπ‘šξ…ž], π‘šβ€²=2, 4, 5, 8, 10, 50.

Remark 2.2. If βŸ¨π‘›βŸ©=Π𝑠𝑖=1πœƒπ‘– in cyclotomic ring β„€[πœπ‘šξ…ž], where πœƒπ‘– is an ideal and 𝑠 is an odd integer, then there is no solution to 𝛿𝛿=𝑛. To see this, assume that a 𝛿 exist such that βŸ¨π›ΏβŸ©=Ξ π‘˜π‘–=1𝛼𝑖. Then βŸ¨π‘›βŸ©=βŸ¨π›ΏβŸ©βŸ¨π›ΏβŸ© has 2π‘˜ factors but βŸ¨π‘›βŸ© has odd factors.

Remark 2.3. Let us consider the ideal generated by 2 which has two factors in the cyclotomic ring β„€[𝜁23]. We claim that the algebraic number 2 is prime in this ring. Since (23, 11, 5) difference sets exist, and there exists πœƒ such that πœƒπœƒ=6 and πœƒ+πœƒ=βˆ’1. This implies that πœƒ2+πœƒ+6=0 and βˆšπœƒ=(βˆ’1Β±βˆ’23)/2. Consequently, βˆšπœƒβˆˆβ„€[βˆ’23]. Suppose that the algebraic number 2 is not prime in βˆšβ„€[βˆ’23]. As βˆ’23≑1(mod4)([23], chapter 3), we seek π‘Ž, π‘βˆˆβ„€ such that βˆšπ›Ώ=(π‘Ž+π‘βˆ’23)/2 and 𝛿𝛿=(π‘Ž2+23𝑏2)/4=2. The equation π‘Ž2+23𝑏2=8 has no integer solution. Thus, there is no algebraic number such that 𝛿𝛿=2. In fact, 𝛿𝛿=𝑝 has no solution, where 𝑝=2, 3, 5, 7, 10, 11, 13, 14, 15, 17, 19, 20, 21, 22. However, the equation π‘Ž2+23𝑏2=4π‘š2 has trivial solutions (π‘Ž,𝑏)=(βˆ’2π‘š,0) and (2π‘š,0), where π‘š=2, 3, 4, 5, 7, 9, 10, 11, 13, 14, 15, 17, 19, 20, 21, 22, 25. We noticed that since the class number of the cyclotomic ring β„€[𝜁23] is 3, the equation π‘Ž2+23𝑏2=4β‹…23 has nontrivial solutions (π‘Ž,𝑏)=(βˆ’3,βˆ’1), (βˆ’3,1), (3,βˆ’1) and (3,1). Also, π‘Ž2+23𝑏2=4π‘š has nontrivial solutions for π‘š=6, 8, 12, 18, 62, 82, 122, 162, 182 and 242.

In this paper, we will use the phase π‘š factors trivially in β„€[πœπ‘šξ…ž] if the ideal generated by π‘š is prime or ramifies in β„€[πœπ‘šξ…ž]; π‘š is self-conjugate modulo π‘šβ€²; the ideal generated by π‘š has odd factors or the algebraic equation 𝛿𝛿=π‘š2 has no solution or has trivial solutions. In summary, suppose that 𝐷 is the difference set image of order 𝑛=π‘š2 in the cyclic factor group 𝐺/𝑁, where 𝐺/𝑁 is a group with exponent π‘šβ€². Suppose that π‘š factors trivially in β„€[πœπ‘šξ…ž] and πœ’ is a nontrivial representation of 𝐺/𝑁. Then ξπœ’(𝐷)=Β±π‘šπœπ‘–π‘šξ…ž, πœπ‘šξ…ž is the π‘šβ€²th root of unity [13].

2.3. Characteristics of Difference Set Images in Subgroup of a Group

In this subsection, we use the attributes of subgroups of a group to obtain information about the difference set image in the subgroups. Dillon [5] proved the following results which will be used to obtain difference set images in dihedral group of a certain order if the difference images in the cyclic group of same order are known.

Theorem 2.4 (dillon dihedral trick). Let 𝐻 be an abelian group and let 𝐺 be the generalized dihedral extension of 𝐻. That is, 𝐺=βŸ¨π‘ž,π»βˆΆπ‘ž2=1,π‘žβ„Žπ‘ž=β„Žβˆ’1,βˆ€β„Žβˆˆπ»βŸ©. If 𝐺 contains a difference set, then so does every abelian group which contains 𝐻 as a subgroup of index 2.

Corollary 2.5. If the cyclic group 𝑍2π‘š does not contain a (nontrivial) difference set, then neither does the dihedral group of order 2π‘š.

Remark 2.6. We look at subgroup properties of a group that can aid the construction of difference set image. For the convenience of the reader, we reproduce the idea of Gjoneski et al. [26]. Suppose that 𝐻 is a group of order 2β„Ž with a central involution 𝑧. We take 𝑇={π‘‘π‘–βˆΆπ‘–=1,…,β„Ž} to be the transversal of βŸ¨π‘§βŸ© in 𝐻 so that every element in 𝐻 is viewed as 𝑑𝑖𝑧𝑗, 0β‰€π‘–β‰€β„Ž, 𝑗=0,1. Denote the set of all integral combinations, βˆ‘β„Žπ‘–=1π‘Žπ‘–π‘‘π‘– of elements of 𝑇, π‘Žπ‘–βˆˆβ„€ by β„€[𝑇]. Using the two representations of subgroup βŸ¨π‘§βŸ© and Frobenius reciprocity theorem [27], we may write any element 𝑋 of the group ring β„€[𝐻] in the form 𝑋=𝑋1+𝑧2+𝑋1βˆ’π‘§2.(2.5) Furthermore, let 𝐴 be the group ring element created by replacing every occurrence of 𝑧 in 𝑋 by 1. Also, let 𝐡 be the group ring element created by replacing every occurrence of 𝑧 in 𝐻 by βˆ’1. Then 𝑋=π΄βŸ¨π‘§βŸ©2ξ‚Άξ‚΅+𝐡2βˆ’βŸ¨π‘§βŸ©2ξ‚Ά,(2.6) where βˆ‘π΄=β„Žπ‘–=1π‘Žπ‘–π‘‘π‘– and βˆ‘π΅=β„Žπ‘—=1𝑏𝑗𝑑𝑗, π‘Žπ‘–, π‘π‘—βˆˆβ„€. As π‘‹βˆˆβ„€[𝐻], 𝐴 and 𝐡 are both in β„€[𝑇], and 𝐴≑𝐡mod2. We may equate 𝐴 with the homomorphic image of 𝑋 in 𝐺/βŸ¨π‘§βŸ©. Consequently, if 𝑋 is a difference set, then the coefficients of 𝑑𝑖 in the expression for 𝐴 will be intersection number of 𝑋 in the coset βŸ¨π‘§βŸ©[26]. In particular, it can be shown that if 𝐾 is a subgroup of a group 𝐻 such that π»β‰…πΎΓ—βŸ¨π‘§βŸ©,(2.7) then the difference set image in 𝐻 is 𝐷=π΄βŸ¨π‘§βŸ©2ξ‚Άξ‚΅+𝑔𝐡2βˆ’βŸ¨π‘§βŸ©2ξ‚Ά,(2.8) where π‘”βˆˆπ», 𝐴 is a difference set in 𝐾, 𝛼=(π‘˜+π‘š)/|𝐾| or 𝛼=(π‘˜βˆ’π‘š)/|𝐾|, 𝐡=π΄βˆ’π›ΌπΎ and π‘˜ is the size of the difference set. Equation (2.8) is true as long as |𝐾|∣(π‘˜+π‘š) or |𝐾|∣(π‘˜βˆ’π‘š).

2.4. Amalgamation of Results

In this paper, we study (𝑣,π‘˜,πœ†) difference sets in which 𝑛=π‘˜βˆ’πœ†=π‘š2 and the ideal generated by π‘š factors trivially in the cyclotomic ring β„€[πœπ‘šξ…ž]. That is, if 𝑛=π‘š2, then (𝑛)=(π‘š)(π‘š) up to units in β„€[πœπ‘šξ…ž]. This method is very useful in the investigation of difference sets in solvable groups. A group 𝐺 is solvable if the sequence πΊβŠ‡πΊξ…žβŠ‡πΊξ…žξ…žβ‹―βŠ‡β‹―βŠ‡πΊ(𝑖)β‹― terminates in the identity, 𝐺(𝑒)=1, in a finite number of steps, each 𝐺(𝑖) is the derived group of the preceding one [28]. Consequently, each 𝑖, the factor group 𝐺(𝑖)/𝐺(𝑖+1) is Abelian. We now state the extended Sylow theorem in solvable groups ([28], page 141).

Theorem 2.7. Let 𝐺 be a solvable group of order π‘šπ‘›, in which gcd(π‘š,𝑛)=1. Then(1)𝐺 possesses at least one subgroup of order π‘š; (2)any two subgroups of order π‘š are conjugates;(3)any subgroup whose order π‘šβ€² divides π‘š is contained in a subgroup of order π‘š; (4)the number π‘›π‘š of subgroups of order π‘š may be expressed as a product of factors, each of which (a) is congruent to 1 modulo some prime factor of π‘š, and (b) is a power of a prime and divides one of the chief factors of 𝐺.

The next three criteria enable us to rule out the existence of difference sets.

Criterion 1. Suppose that 𝐺 is a group of order 𝑣=𝑝′𝑠, where 𝑝′ is prime, 𝑠 and 𝑑 are integers. Then 𝐺 does not admit (𝑣,π‘˜,πœ†) if there exists a normal subgroup 𝑁 of 𝐺 such that(1)π‘˜βˆ’πœ†=π‘š2, π‘š is a natural number, (2)|𝐺/𝑁|=𝑝′, (3)π‘š factors trivially in the cyclotomic ring β„€[πœπ‘ξ…ž], where πœπ‘ξ…ž is the 𝑝′th root of unity, (4)the difference set solution in 𝐺/𝑁 is one of the forms 𝛼(𝐺/𝑁)+π‘š, 𝛼+π‘š>|𝑁| or 𝛼(𝐺/𝑁)βˆ’π‘š, 𝛼<π‘š.

Proof. The nonexistence of viable difference set image in 𝐺/𝑁 implies that 𝐺 does not admit (𝑣,π‘˜,πœ†) difference set.

In this criterion, we may replace |𝐺/𝑁|=𝑝′ with |𝐺/𝑁|=π‘β€²π‘ž if π‘ž>2 is prime power, π‘žβˆ£π‘ , gcd(𝑝′,π‘ž)=1, and the ideal generated by 𝑝 factors trivially in β„€[πœπ‘ξ…žπ‘ž], where 𝑝 is a prime divisor of π‘š (see Remark 2.3).

Criterion 2. Suppose that 𝐺 is a group of even order 𝑣 and 𝐻 is a factor group of 𝐺 with |𝐻|=2π‘ž, where π‘ž is prime. Let 𝑔 be an element of order 2 in 𝐻. Then 𝐺 does not admit (𝑣,π‘˜,πœ†) if(1)π‘˜βˆ’πœ†=π‘š2, π‘š is a natural number, (2)π‘š factors trivially in the cyclotomic rings β„€[πœπ‘ž], where πœπ‘ž is π‘žth root of unity, (3)the difference set solution in 𝐻/βŸ¨π‘”βŸ© is one of the forms 𝛼(𝐻/βŸ¨π‘”βŸ©)+π‘š,  𝛼+π‘š>|𝐺/(𝐻/βŸ¨π‘”βŸ©)| or 𝛼(𝐻/βŸ¨π‘”βŸ©)βˆ’π‘š,  𝛼<π‘š; alternatively, the difference set image in 𝐻 is one of the forms 𝛼(𝐻)+π‘š,  𝛼+π‘š>|𝐺/𝐻| or 𝛼(𝐻)βˆ’π‘š,  𝛼<π‘š.

Proof. The proof follows from Criterion 1 and the fact that if |𝐻|≑2(mod4), the cyclotomic rings β„€[𝜁2π‘ž] and β„€[πœπ‘ž] are the same.

Criterion 3. Suppose that 𝐺 is a group of order 𝑣=22Γ—π‘žΓ—π‘ , where π‘žβ‰₯3 is prime and 𝑠 is an integer. Suppose that 𝐻 is a factor group of 𝐺 of order 2π‘ž. The group 𝐺 does not admit (𝑣,π‘˜,πœ†) difference set if there exists a normal subgroup 𝑁 such that 𝐺/𝑁≅𝐻×𝐢2 and(1)π‘˜βˆ’πœ†=π‘š2,β€‰β€‰π‘š is a natural number, (2)every prime divisor π‘šβ€² of π‘š factors trivially in the cyclotomic rings β„€[πœπ‘ž], where πœπ‘ž is π‘žth root of unity, gcd(π‘šβ€²,π‘ž)=1(3)the difference set solution in 𝐻 is of the form 𝛼𝐻+π‘š, and 𝛼 is an odd integer or (4)the difference set solution in 𝐻 is of the form π›Όπ»βˆ’π‘š, 𝛼 is an even integer, π‘š is an odd integer, and π‘š>𝛼/2.

Proof. There are two groups of order 2π‘ž, cyclic and dihedral groups. Since every prime divisor π‘šβ€² of π‘š factors trivially in the cyclotomic rings β„€[πœπ‘ž], gcd(π‘šβ€²,π‘ž)=1, it follows that π‘š factors trivially in β„€[πœπ‘ž]. Consequently, the (𝑣,π‘˜,πœ†) difference set image in 𝐻 is of the form 𝛼𝐻+π‘š or π›Όπ»βˆ’π‘š. Suppose that the (𝑣,π‘˜,πœ†) difference set image in 𝐻 is of the form 𝛼𝐻+π‘š and 𝛼 is an odd integer. Using (2.8), the difference set image in 𝐻×𝐢2 is 𝐷=π΄βŸ¨π‘§βŸ©2ξ‚Άξ‚΅+𝑔𝐡2βˆ’βŸ¨π‘§βŸ©2ξ‚Ά,(2.9) where π‘”βˆˆπ»Γ—πΆ2, 𝐴=𝛼𝐻+π‘š is the difference set image in 𝐻, 𝐡=π΄βˆ’π›Όπ» with 𝛼=(π‘˜+π‘š)/2π‘ž or (π‘˜βˆ’π‘š)/2π‘ž and 𝑧 is the generator of 𝐢2. Since 𝛼 is odd, 𝐴(βŸ¨π‘§βŸ©/2) consists of at least 2π‘žβˆ’2 odd entries while 𝐡((2βˆ’βŸ¨π‘§βŸ©)/2) consists of at least 2π‘žβˆ’2 even entries. Thus, (2.8) has no integer solutions. On the other hand, suppose that the (𝑣,π‘˜,πœ†) difference set image in 𝐻 is of the form π›Όπ»βˆ’π‘š, 𝛼 is an even integer, π‘š is odd an odd integer, and π‘š>𝛼/2. The difference set image is of the form (2.9) with 𝐴=π›Όπ»βˆ’π‘š. Since 𝛼 is even and π‘š is odd, 𝐴(βŸ¨π‘§βŸ©/2) and 𝐡((2βˆ’βŸ¨π‘§βŸ©)/2) have two entries that are fractions. In particular, we can translate if necessary, to ensure that the coefficients of the identity in both components are (π›Όβˆ’π‘š)/2 and βˆ’π‘š/2, respectively. The sum and difference of these two entries are, respectively, (π›Όβˆ’2π‘š)/2 and 𝛼/2. But π‘š>𝛼/2 and 2π‘š>𝛼. Hence, (π›Όβˆ’2π‘š)/2 is a negative integer. Thus, there is no difference set image in 𝐻×𝐢2 and the criterion follows.

Notice that there are five factor groups of order 22Γ—π‘ž if π‘žβ‰‘1(mod4) and four factor groups if π‘žβ‰‘3(mod4). Criterion 3 rules out the existence of difference set images in πΆπ‘žΓ—πΆ2×𝐢2 and 𝐷2π‘žβ‰…π·π‘žΓ—πΆ2. In addition to conditions of Criterion 3, if π‘š factors trivially also in β„€[𝜁22Γ—π‘ž], then three of the four or five factor groups (𝐢2π‘ž, πΆπ‘žΓ—πΆ2×𝐢2 and 𝐷2π‘ž) of order 22Γ—π‘ž do not admit difference sets.

3. Some Difference Sets Parameters (Tables 1–5)

tab1
Table 1: Parameter sets that do not exist by Criterion 1. 𝐢|𝐺/𝑁|=⟨π‘₯⟩ and parameters with asterisk indicate new results.
tab2
Table 2: Partial results in groups of order 𝑣 by Criterion 1. 𝐢|𝐺/𝑁|=⟨π‘₯⟩ and parameters with asterisk indicate new results. ? means the number of groups of order 𝑣 is unknown.
tab3
Table 3: Parameter sets that do not exist by Criterion 2. 𝐢|𝐻|=⟨π‘₯,π‘¦βˆΆπ‘₯π‘ž=𝑦2=[π‘₯,𝑦]⟩ and parameters with asterisk indicate new results. No diff. set image in 𝐻 implies no difference set image in Dihedral group of same order.
tab4
Table 4: Partial results in groups of order 𝑣 by Criterion 2. 𝐢|𝐻|=⟨π‘₯,π‘¦βˆΆπ‘₯π‘ž=𝑦2=[π‘₯,𝑦]⟩ and parameters with asterisk indicate new results. No difference set image in 𝐻 implies no diff. set image in Dihedral group of same order. ? means the number of groups of order 𝑣 is unknown.
tab5
Table 5: Partial results in groups of order 𝑣 by Criterion 3. Parameters with asterisk indicate new results.? means the number of groups of order 𝑣 is unknown.

We list some parameter sets (both known and new) that do not exist. In each of these cases, 𝐺 is a group of order 𝑣 and πœ‘βˆΆπΊβ†’π» is a group homomorphism. Suppose that 𝐷 is a π‘˜-subset of 𝐺 and 𝑛=π‘˜βˆ’πœ†=π‘š2 such that π‘š factors trivially in the cyclotomic ring β„€[𝜁|𝐻|]. We use Criteria 1, 2 and 3 to rule out the existence of (𝑣,π‘˜,πœ†) difference set. Examples of such parameters are listed in Tables 1 and 3. We also listed partial results in Tables 2, 4, and 5.

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