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Volume 2012 (2012), Article ID 367129, 19 pages
On the Existence of Difference Sets with and 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. Kelarev, D. Kressner, H. You, and A. Zimmermann
Copyright © 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.
We combine Turyn's self-conjugacy result, variance technique, Dillon dihedral trick, and Sylow theorem to investigate the existence of () difference sets in which is a square and .
Let be a multiplicative group of order and let be a subset of consisting of elements, where . is a nontrivial difference set if every nonidentity element can be reproduced times by the multiset . The natural number 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 is abelian (resp., nonabelian or cyclic), then 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 . 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  or Turyn’s self-conjugacy method . Recently, Schmidt  developed a new method for studying combinatorial structures using group ring equations without any restrictive assumptions. Arasu , Arasu and Sehgal [3, 16], Baumert , Hughes , Iiams , Kibler , Kopilovich , Lander , and López and Sánchez  among other authors studied the existence of abelian difference sets with . 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 difference sets in which , where is a positive integer. We illustrate with examples where and the ideal generated by prime divisors of 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 .
We assume that is a finite group of order . 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 .
2.1. Difference Sets
Let be the ring of integers and be the field of complex numbers. Suppose that is a group of order and is a difference set in . We sometimes view the elements of as members of the group ring , which is a subring of the group algebra . Thus, represents both subset of and element of . The sum of inverses of elements of is . Consequently, is a difference set if and only if Suppose that is a difference set in a group of order and is a normal subgroup of . Suppose that is a homomorphism. We can extend by linearity, to the corresponding group rings. Thus, the difference set image in (also known as the contraction of with respect to the kernel ) is the multiset . Let be a left transversal of in . We can write , where the integer is known as the intersection number of with respect to . In this work, we will always use the notation for and denotes the number of times equals .
2.2. Representation and Algebraic Number Theories
A -representation of is a homomorphism, , where is the group of invertible matrices over . The positive integer is the degree of . A linear representation (character) is a representation of degree one. The set of all linear representations of is denoted by . is an abelian group under multiplication and if is the derived group of , then is isomorphic to . Define to be a primitive th root of unity and to be the cyclotomic extension of the field of rational numbers, , where is the exponent of . Without loss of generality, we may replace by the field . Thus, the central primitive idempotents in is where is an irreducible character of .
Aliases are members of group ring which enable us to transfer information from to group algebra and then to . Let be an abelian group and let , be the set of characters of . The element is known as -alias if for and all , . Since , we can replace the occurrence of , which is a complex number by -alias, , an element of . Furthermore, two characters of are algebraic conjugate if and only if they have the same kernel and we denote the set of equivalence classes of by . The central rational idempotents in are obtained by summing over the equivalence classes on the 's under the action of the Galois group of over . That is, , .
For instance, suppose ( is prime) is a cyclic group whose characters are of the form , . Then the rational idempotents are The following is the general formula employed in the search of difference set .
Theorem 2.1. Let be an abelian group and be the set of equivalence classes of characters. Suppose that is a system of distinct representatives for the equivalence classes of . Then for , one has where is any -alias for .
Equation (2.4) is known as the rational idempotent decomposition of .
Suppose that is any nontrivial representation of degree and , where is the primitive root of unity. Suppose that is a nonidentity element. Then, . This shows that . Since is not an identity element, and ( is an integral domain). Consequently, , where is the identity matrix. Furthermore, if is a nontrivial representation of of degree then and .
Recall that the ring of integers of the cyclotomic field is . This ring is also an integral domain. Let , , . The number is irreducible if implies one of or is a unit. The element is prime if implies or . 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 is an abelian factor group of exponent and is a difference set image in . If is not a principal character of , then is an algebraic equation in . The determination of the alias requires the knowledge of how the ideal generated by factors in cyclotomic ring , where is the th root of unity. If , then by (2.4), we seek such that . The task of solving the algebraic equation is sometimes made easier if we consider the factorization of principal ideals . Suppose we are able to find such that , where is the Euler -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 , where is a unit.
The following result is used to determine the number of factors of an ideal in a ring: suppose is any prime and is an integer such that . Suppose that is the order of in the multiplicative group of the modular number ring . Then the number of prime ideal factors of the principal ideal in the cyclotomic integer ring is , where is the Euler -function, that is, . For instance, the ideal generated by 2 has two factors in , the ideal generated by 7 has two factors in , while the ideal generated by 3 has four factors in . On the other hand, since is a power of 2, the ideal generated by 2 is said to completely ramifies as power of in .
According to Turyn , an integer is said to be semi-primitive modulo if for every prime factor of , there is an integer such that . In this case, −1 belongs to the multiplicative group generated by . Furthermore, is self-cosnjugate modulo if every prime divisor of is semi primitive modulo , where is the largest divisor of relatively prime to . This means that all prime ideals over in are fixed by complex conjugation. For instance, , where , 5, 10 and , , 4, 8. Thus, is fixed by conjugation in , , 4, 5, 8, 10, 50.
Remark 2.2. If in cyclotomic ring , where is an ideal and is an odd integer, then there is no solution to . To see this, assume that a exist such that . Then has factors but has odd factors.
Remark 2.3. Let us consider the ideal generated by 2 which has two factors in the cyclotomic ring . We claim that the algebraic number 2 is prime in this ring. Since (23, 11, 5) difference sets exist, and there exists such that and . This implies that and . Consequently, . Suppose that the algebraic number 2 is not prime in . As (, chapter 3), we seek , such that and . The equation has no integer solution. Thus, there is no algebraic number such that . In fact, has no solution, where , 3, 5, 7, 10, 11, 13, 14, 15, 17, 19, 20, 21, 22. However, the equation has trivial solutions and , where , 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 is 3, the equation has nontrivial solutions , , and . Also, has nontrivial solutions for , 8, 12, 18, 62, 82, 122, 162, 182 and 242.
In this paper, we will use the phase factors trivially in if the ideal generated by is prime or ramifies in ; is self-conjugate modulo ; the ideal generated by has odd factors or the algebraic equation has no solution or has trivial solutions. In summary, suppose that is the difference set image of order in the cyclic factor group , where is a group with exponent . Suppose that factors trivially in and is a nontrivial representation of . Then , is the th root of unity .
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  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 be an abelian group and let be the generalized dihedral extension of . That is, . If contains a difference set, then so does every abelian group which contains as a subgroup of index 2.
Corollary 2.5. If the cyclic group does not contain a (nontrivial) difference set, then neither does the dihedral group of order .
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. . Suppose that is a group of order with a central involution . We take to be the transversal of in so that every element in is viewed as , , . Denote the set of all integral combinations, of elements of , by . Using the two representations of subgroup and Frobenius reciprocity theorem , we may write any element of the group ring in the form Furthermore, let be the group ring element created by replacing every occurrence of in by 1. Also, let be the group ring element created by replacing every occurrence of in by −1. Then where and , , . As , and are both in , and . We may equate with the homomorphic image of in . Consequently, if is a difference set, then the coefficients of in the expression for will be intersection number of in the coset . In particular, it can be shown that if is a subgroup of a group such that then the difference set image in is where , is a difference set in , or , and is the size of the difference set. Equation (2.8) is true as long as or .
2.4. Amalgamation of Results
In this paper, we study difference sets in which and the ideal generated by factors trivially in the cyclotomic ring . That is, if , then up to units in . This method is very useful in the investigation of difference sets in solvable groups. A group is solvable if the sequence terminates in the identity, , in a finite number of steps, each is the derived group of the preceding one . Consequently, each , the factor group is Abelian. We now state the extended Sylow theorem in solvable groups (, page 141).
Theorem 2.7. Let be a solvable group of order , in which . Then(1) possesses at least one subgroup of order ; (2)any two subgroups of order are conjugates;(3)any subgroup whose order divides is contained in a subgroup of order ; (4)the number of subgroups of order may be expressed as a product of factors, each of which (a) is congruent to 1 modulo some prime factor of , and (b) is a power of a prime and divides one of the chief factors of .
The next three criteria enable us to rule out the existence of difference sets.
Criterion 1. Suppose that is a group of order , where is prime, and are integers. Then does not admit if there exists a normal subgroup of such that(1), is a natural number, (2), (3) factors trivially in the cyclotomic ring , where is the th root of unity, (4)the difference set solution in is one of the forms , or , .
Proof. The nonexistence of viable difference set image in implies that does not admit difference set.
In this criterion, we may replace with if is prime power, , , and the ideal generated by factors trivially in , where is a prime divisor of (see Remark 2.3).
Criterion 2. Suppose that is a group of even order and is a factor group of with , where is prime. Let be an element of order 2 in . Then does not admit if(1), is a natural number, (2) factors trivially in the cyclotomic rings , where is th root of unity, (3)the difference set solution in is one of the forms , or , ; alternatively, the difference set image in is one of the forms , or , .
Proof. The proof follows from Criterion 1 and the fact that if , the cyclotomic rings and are the same.
Criterion 3. Suppose that is a group of order , where is prime and is an integer. Suppose that is a factor group of of order . The group does not admit difference set if there exists a normal subgroup such that and(1), is a natural number, (2)every prime divisor of factors trivially in the cyclotomic rings , where is th root of unity, (3)the difference set solution in is of the form , and is an odd integer or (4)the difference set solution in is of the form , is an even integer, is an odd integer, and .
Proof. There are two groups of order , cyclic and dihedral groups. Since every prime divisor of factors trivially in the cyclotomic rings , , it follows that factors trivially in . Consequently, the difference set image in is of the form or . Suppose that the difference set image in is of the form and is an odd integer. Using (2.8), the difference set image in is where , is the difference set image in , with or and is the generator of . Since is odd, consists of at least odd entries while consists of at least even entries. Thus, (2.8) has no integer solutions. On the other hand, suppose that the difference set image in is of the form , is an even integer, is odd an odd integer, and . The difference set image is of the form (2.9) with . Since is even and is odd, and 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 and , respectively. The sum and difference of these two entries are, respectively, and . But and . Hence, is a negative integer. Thus, there is no difference set image in and the criterion follows.
Notice that there are five factor groups of order if and four factor groups if . Criterion 3 rules out the existence of difference set images in and . In addition to conditions of Criterion 3, if factors trivially also in , then three of the four or five factor groups (, and ) of order do not admit difference sets.
We list some parameter sets (both known and new) that do not exist. In each of these cases, is a group of order and is a group homomorphism. Suppose that is a -subset of and such that factors trivially in the cyclotomic ring . We use Criteria 1, 2 and 3 to rule out the existence of 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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