A Construction of Difference Sets in High Exponent 2-Groups Using Representation Theory

Nontrivial difference sets in groups of order a power of 2 are part of the family of difference sets called Menon difference sets (or Hadamard), and they have parameters (2^2d+2, 2^2d+1±2 ^d , 2^2d ±2 ^d ). In the abelian case, the group has a difference set if and only if the exponent of the group is less than or equal to 2 ^d+2. In [14], the authors construct a difference set in a nonabelian group of order 64 and exponent 32. This paper generalizes that result to show that there is a difference set in a nonabelian group of order 2^2d+2 with exponent 2 ^d+3. We use representation theory to prove that the group has a difference set, and this shows that representation theory can be used to verify a construction similar to the use of character theory in the abelian case.     

Indivisible plexes in latin squares

In this paper, we present two constructions of divisible difference sets based on skew Hadamard difference sets. A special class of Hadamard difference sets, which can be derived from a skew Hadamard difference set and a Paley type regular partial difference set respectively in two groups of orders v ₁ and v ₂ with |v ₁ − v ₂| = 2, is contained in these constructions. Some result on inequivalence of skew Hadamard difference sets is also given in the paper. As a consequence of Delsarte’s theorem, the dual set of skew Hadamard difference set is also a skew Hadamard difference set in an abelian group. We show that there are seven pairwisely inequivalent skew Hadamard difference sets in the elementary abelian group of order 3⁵ or 3⁷, and also at least four pairwisely inequivalent skew Hadamard difference sets in the elementary abelian group of order 3⁹. Furthermore, the skew Hadamard difference sets deduced by Ree-Tits slice symplectic spreads are the dual sets of each other when q ≤ 3¹¹.      

A New Family of Relative Difference Sets in 2-Groups

We recursively construct a new family of ( 2^6d+4, 8, 2^6d+4, 2^6d+1) semi-regular relative difference sets in abelian groups G relative to an elementary abelian subgroup U. The initial case d = 0 of the recursion comprises examples of (16, 8, 16, 2) relative difference sets for four distinct pairs (G, U).     

New negative Latin square type partial difference sets in nonelementary abelian 2-groups and 3-groups

A partial difference set having parameters (n ², r(n − 1), n + r ² − 3r, r ² − r) is called a Latin square type partial difference set, while a partial difference set having parameters (n ², r(n + 1), − n + r ² + 3r, r ² + r) is called a negative Latin square type partial difference set. Nearly all known constructions of negative Latin square partial difference sets are in elementary abelian groups. In this paper, we develop three product theorems that construct negative Latin square type partial difference sets and Latin square type partial difference sets in direct products of abelian groups G and G′ when these groups have certain Latin square or negative Latin square type partial difference sets. Using these product theorems, we can construct negative Latin square type partial difference sets in groups of the form where the s _i are nonnegative integers and s ₀ + s ₁ ≥ 1. Another significant corollary to these theorems are constructions of two infinite families of negative Latin square type partial difference sets in 3-groups of the form for nonnegative integers s _i . Several constructions of Latin square type PDSs are also given in p-groups for all primes p. We will then briefly indicate how some of these results relate to amorphic association schemes. In particular, we construct amorphic association schemes with 4 classes using negative Latin square type graphs in many nonelementary abelian 2-groups; we also use negative Latin square type graphs whose underlying sets can be elementary abelian 3-groups or nonelementary abelian 3-groups to form 3-class amorphic association schemes.      

A Note on Certain 2-Groups with Hadamard Difference Sets

Nontrivial difference sets in 2-groups are part of the family of Hadamarddifference sets. An abelian group of order 2^2d+2 has a difference setif and only if the exponent of the group is less than or equal to2 ^d+2. We provide an exponent bound for a more general type of 2-groupwhich has a Hadamard difference set. A recent construction due to Davis and Iiamsshows that we can attain this bound in at least half of the cases.     

Straightening and bounded cohomology of hyperbolic groups

It was stated by M. Gromov [Gr2] that, for any hyperbolic group G, the map from bounded cohomology Hⁿ_b(G,\Bbb R) H^n_b(G,{\Bbb R}) to Hⁿ(G,\Bbb R) H^n(G,{\Bbb R}) induced by inclusion is surjective for n 3 2 n \ge 2 . We introduce a homological analogue of straightening simplices, which works for any hyperbolic group. This implies that the map Hⁿ_b(G,V) ? Hⁿ(G,V) H^n_b(G,V) \to H^n(G,V) is surjective for n 3 2 n \ge 2 when V is any bounded \Bbb QG {\Bbb Q}G -module and when V is any finitely generated abelian group.     

Exponential number of inequivalent difference sets in ℤ

Kantor [ 5 ] proved an exponential lower bound on the number of pairwise inequivalent difference sets in the elementary abelian group of order 2^2s+2. Dillon [ 3 ] generalized a technique of McFarland [ 6 ] to provide a framework for determining the number of inequivalent difference sets in 2‐groups with a large elementary abelian direct factor. In this paper, we consider the opposite end of the spectrum, the rank 2 group ? , and compute an exponential lower bound on the number of pairwise inequivalent difference sets in this group. In the process, we demonstrate that Dillon difference sets in groups ? can be constructed via the recursive construction from [ 2 ] and we show that there are exponentially many pairwise inequivalent difference sets that are inequivalent to any Dillon difference set. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 249–259, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10046     

Reversible and DRAD difference sets in

The existence of Hadamard difference sets has been a central question in design theory. Reversible difference sets have been studied extensively. Dillon gave a method for finding reversible difference sets in groups of the form (C)². DRAD difference sets are a newer concept. Davis and Polhill showed the existence of DRAD difference sets in the same groups as Dillon. This article determines the existence of reversible and DRAD difference sets in groups of the form (C)³. These are the only abelian 2‐groups outside of direct products of C₄ and (C)² known to contain reversible and DRAD difference sets. © 2011 Wiley Periodicals, Inc. J Combin Designs 20:58–67, 2012     

On matrix near-rings

Let R be a right near-ring with identity and M_n(R) be the near-ring of n 2 n matrices over R in the sense of Meldrum and Van der Walt. In this paper, M_n(R) is said to be s\sigma-generated if every n 2 n matrix A over R can be expressed as a sum of elements of X_n(R), where X_n(R)={f_ij^r | 1\leqq i, j\leqq n, r ? R}X_n(R)=\{f_{ij}^r\,|\,1\leqq i, j\leqq n, r\in R\}, is the generating set of M_n(R). We say that R is s\sigma-generated if M_n(R) is s\sigma-generated for every natural number n. The class of s\sigma-generated near-rings contains distributively generated and abstract affine near-rings. It is shown that this class admits homomorphic images. For abelian near-rings R, we prove that the zerosymmetric part of R is a ring, so the class of zerosymmetric abelian s\sigma-generated near-rings coincides with the class of rings. Further, for every n, there is a bijection between the two-sided subgroups of R and those of M_n(R).     

On the equivariant structure of ideals in abelian extensions of local fields (with an appendix by W. Bley)

Let p be an odd rational prime and K a finite extension of \Bbb Q_p {\Bbb Q}_p . We give a complete classification of those finite abelian extensions L/K L/K in which any ideal of the valuation ring of L is free over its associated order in \Bbb Q_p[Gal(L/K)] {\Bbb Q}_p[Gal(L/K)] . In an appendix W. Bley describes an algorithm which can be used to determine the structure of Galois stable ideals in abelian extensions of number fields. The algorithm is applied to give several new and interesting examples.     

Varietal isologism and covering groups

In this paper it will be shown that any two \bf\cal V-covering groups of a given group are V\bf\cal V-isologic with respect to the variety V\bf\cal V, which is a vast generalization of a result in B. Huppert (1967) and R. L. Griess JR (1973). We also give a criterion of existence of V\bf\cal V-covering groups for a V\bf\cal V-perfect group, and show that every automorphism of a given V\bf\cal V-perfect group G can be extended to an automorphism of the V\bf\cal V-covering G^* say, of G, this generalizes a result of J. L. Alperin and D. Gorenstein (1966), in the abelian variety.     

Using rational idempotents to show Turyn’s bound is sharp

Davis, Dillon, and Jedwab all showed the existence of difference sets in groups C_2^r+2×C_2^r{C_{2^{r+2}}\times C_{2^{r}}} . Turyn’s bound had previously shown that abelian 2-groups with higher exponents could not admit difference sets. We give a new construction technique that utilizes character values, rational idempotents, and tiling structures to produce Hadamard difference sets in the group C_2^r+2×C_2^r{C_{2^{r+2}}\times C_{2^{r}}} to replicate the result.     

On the “three-space” problem for locally quasi-convex topological groups

We study, in the context of abelian topological groups, the "three-space" problem for the property of being locally quasi-convex, after a paper of M. Bruguera. Our main contributions are: establishing a 3-lemma suitable to work with topological groups (which allows to translate the basic elements of homological algebra to the category of topological groups) and obtaining the analogue, for topological groups, of Dierolf's result in topological vector spaces:¶Theorem. Given two abelian locally quasi-convex groups H and G there exists a non-locally-quasi-convex extension of H and G if and only if there exists a non-locally-quasi-convex extension of S (the circle group) and G.     

On finite quasi-isometric sets in $ \mathbb{R}^n $

We compare two concepts from distance geometry of finite sets: quasi-isometry and isometry. We show that for every n 3 5 n\geq5 there exist sets of n points in \mathbbR^n-1 \mathbb{R}^{n-1} that are quasi-isometric and not isometric. By contrast, for finite sets in S¹ we show that under some additional hypotheses, quasi-isometric sets are isometric.     

Abelian Kernels of Some Monoids of Injective Partial Transformations and an Application

In this paper we compute the abelian kernels of the monoids POI_n and POPI_n of all injective order preserving and respectively, orientation preserving, partial transformations on a chain with n elements. As an application, we show that the pseudovariety POPI generated by the monoids POPI_n (n epsilon N) is not contained in the Mal'cev product of the pseudovariety POI generated by the monoids POI_n (n epsilon N) with the pseudovariety Ab of all finite abelian groups.     

Group Connectivity of 3-Edge-Connected Chordal Graphs

Let A be a finite abelian group and G be a digraph. The boundary of a function f: E(G)ZA is a function ‘f: V(G)ZA given by ‘f(v)=~_{e leaving v}f(e)m~_{e entering v}f(e). The graph G is A-connected if for every b: V(G)ZA with ~_{v] V(G)} b(v)=0, there is a function f: E(G)ZA{0} such that ‘f=b. In [J. Combinatorial Theory, Ser. B 56 (1992) 165-182], Jaeger et al showed that every 3-edge-connected graph is A-connected, for every abelian group A with |A|̈́. It is conjectured that every 3-edge-connected graph is A-connected, for every abelian group A with |A|̓ and that every 5-edge-connected graph is A-connected, for every abelian group A with |A|́.¶ In this note, we investigate the group connectivity of 3-edge-connected chordal graphs and characterize 3-edge-connected chordal graphs that are A-connected for every finite abelian group A with |A|́.     

On certain dense sets of rationals, simultaneous Schröder and Böttcher equations, and characterizations of functions

Summary. The fact that rational numbers of the forms 2^-m3ⁿ, m and n integers, are dense in the set \mathbbR⁺ \mathbb{R}^+ of non-negative real numbers is crucial in determining well-behaved solutions of a key functional equation. A principal aim of this paper is the presentation of a new proof of the statement that many similar sets of rationals are dense in \mathbbR⁺ \mathbb{R}^+ . The reason for giving a new proof of this statement is that the "standard" argument uses all the basic properties of logarithms and exponentials. The new proof does not, which means that our result can be used without circularity not only in the characterization, but in the very definition of logarithms and exponential functions.     

A family of C-partitions and T-matrices

In this article we give the definition of C-partitions in an abelian group, consider the relation between C-partitions, supplementary difference sets and T-matrices, and for an abelian group of order v = q² with q ≡ 3(mod8) a prime power obtain some constructions of C-partitions and T-matrices. © 1999 John Wiley & Sons, Inc. J Combin Designs 7: 269–281, 1999     

Foldings of Difference Sets in Abelian Groups

 In this paper, we show that under some conditions the existence of a difference set in G implies the existence of another difference set with the same parameters in G′, where G and G′ are abelian groups of the same order. This explains why there are more difference sets in abelian groups of low exponent and high rank than in those of high exponent and low rank. Received: September 1, 1997 / Revised: March 24, 1998     

Difference Sets Corresponding to a Class of Symmetric Designs

We study difference sets with parameters(v, k, ) = (p ^s(r ^2m - 1)/(r - 1), p ^s-1 r ^2m-2 r - 1)r ^{2m -2}, where r = r ^s - 1)/(p - 1) and p is a prime. Examples for such difference sets are known from a construction of McFarland which works for m = 1 and all p,s. We will prove a structural theorem on difference sets with the above parameters; it will include the result, that under the self-conjugacy assumption McFarland's construction yields all difference sets in the underlying groups. We also show that no abelian .160; 54; 18/-difference set exists. Finally, we give a new nonexistence prove of (189, 48, 12)-difference sets in Z ₃ × Z ₉ × Z ₇.