1-Saturating Sets, Caps, and Doubling-Critical Sets in Binary Spaces

Given an additive group G and nonempty subsets S, T of G, let S+T denote the set ﹛s + t | s ∊ S, t ∊ T﹜, S the complement of S in G and |S| the cardinality of S. We call S a sum-free set in G if (S+S) ⊆ S. If, in addition, |S| ≥ |T| for every sum-free set T in G, then we call S a maximal sum-free set in G. We denote by λ(G) the cardinality of a maximal sum-free set in G.

Recently, the concept of (normal) bent partitions, which are partitions of elementary abelian groups having similar properties to spreads, has been introduced by Anbar and Meidl. A large number of bent partitions, so-called generalized semifield spreads, can be obtained from semifields with certain properties. A strongly related concept, namely Latin square partial difference set packings (LP-packings) in finite abelian groups, has also been introduced recently by Jedwab and Li. The examples for LP-packings in an elementary abelian group are obtained from spreads. LP-packings yield bent partitions (not only for elementary abelian groups). In this paper, we first point out that conversely, generalized semifield spreads yield LP-packings. As a result, there is a huge amount of LP-packings in elementary abelian groups, other than spreads. With some examples from ternary bent functions, we then show that normal bent partitions and LP-packings are not the same concept. Finally, we extend the lifting procedure from spreads to LP-packings in nonelementary abelian groups to a lifting procedure from some generalized spreads to LP-packings in nonelementary abelian groups and in larger elementary abelian groups. This potentially yields bent partitions other than generalized semifield spreads.

Maximal sum-free sets in finite Abelian groups, III

Equivalence plays a key-role for the classification of functions between elementary abelian groups $ {{\mathbb V}}_n^{(p)} $ and $ {{\mathbb V}}_k^{(p)} $. One distinguishes between affine equivalence, extended affine (EA) equivalence, and the most general CCZ-equivalence. Recently, there has been an increased interest in functions from elementary abelian groups $ {{\mathbb V}}_n^{(p)} $ to cyclic groups $ {{\mathbb Z}}_{p^k} $. We initiate the study of equivalence for functions from $ {{\mathbb V}}_n^{(p)} $ to $ {{\mathbb Z}}_{p^k} $. We show that CCZ-equivalence is more general than EA-equivalence. For some classes of functions, CCZ-equivalence reduces to EA-equivalence. We show that CCZ-equivalence between two functions from $ {{\mathbb V}}_n^{(p)} $ to $ {{\mathbb Z}}_{p^k} $ implies CCZ-equivalence of two associated vectorial functions from $ {{\mathbb V}}_n^{(p)} $ to $ {{\mathbb V}}_k^{(p)} $.

Bieberbach groups are torsion free crystallographic groups. In this paper, our focus is on the Bieberbach groups with elementary abelian 2-group point group, The central subgroup of the nonabelian tensor square of a group is generated by for all in The purpose of this paper is to compute the central subgroups of the nonabelian tensor squares of two Bieberbach groups with elementary abelian 2-point group of dimension three.

We classify the Morita equivalence classes of principal blocks with elementary abelian defect groups of order 64 with respect to a complete discrete valuation ring with an algebraically closed residue field of characteristic two.

Let K be a quadratic extension field of the rational numbers Q. Let C K be the 2-class group of K in the narrow sense. It is a classical result that rank C K= t l , where t is the number of primes that ramify in K/Q. Now let C~ = {ai: a e CK}, and let R K denote the 4-class rank of K in the narrow sense; i.e., RK= rank ~Kc z = dimv2 ttc'Z/t~41~K/~KJ" Here F 2 is the finite field with two elements, and 2 4 CK/C K is an elementary abelian 2-group which we are viewing as a vector space o v e r F 2. Given a quadratic field K, one can compute R K by computing the rank (over F2) of a certain matrix of Legendre symbols (cf. [11]). Now assume K is imaginary quadratic. So K=Q(1/~mm), where m is a square-free positive integer. For each positive integer t, each nonnegative integer e, and each positive real number x, we define

Elementary proof that [formula omitted] is a DCI-group

We define the positive diameter of a finite group G with respect to a generating set A ⊆ G to be the smallest non-negative integer n such that every element of G can be written as a product of at most n elements of A. This invariant, which we denote by , can be interpreted as the diameter of the Cayley digraph induced by A on G.

Let A be a subset of a finite abelian group G. We say that A is sum-free if there is no solution of the equation x+ y = z, with x, y, z belonging to the set A. Let SF (G) the sel of all sum-free subets of G and σ(G) deonotes the number n(log 2 |SF (G)|). In this article we shall improve the error term in asymptotic formula of σ(G) obtained in [GR05]. The methods used are a slight refinement of methods of [GR05]. Let G be a finite abelian group of order n. A subset A of G is said to be sum-free if there is no solution of the equation x+ y = z, with x, y, z belonging to the set A. Let SF (G) denotes the set of all sum-free subsets of G. This article is motivated by the question of studying the cardinality of the set SF (G). Definition: (I) Let μ(G) denotes the density of a largest sum-free subset of G, so that any such subset has size μ(G)n. (II) Given a set B ⊂ G we say that (x, y, z) ∈ B3 is a Schur triple of the set B if x+ y = z. Observing that all subsets of a sum-free set are sum-free we have the obvious inequality |SF (G)| ≥ 2 (1) Let the symbol σ(G) denotes the number n(log2 |SF (G)|). Then from (1) it follows trivially that σ(G) ≥ μ(G). In this article we improve the results of Ben Green and Imre Ruzsa [GR05] and prove the following two results. The theorem 1 follows immediately from theorem 2 and a result from [GR05], namely theorem 6. The methods used to prove theorem 2 are a slight refinements of methods in [GR05].

Finite Groups Admitting Automorphisms Whose Fixed Point Set is an Elementary Abelian 2-Group

Perfect Isometries for Principal Blocks with Abelian Defect Groups and Elementary Abelian 2-Inertial Quotients

For $2 \leqq c \leqq \infty $, the c-linear complexity $L_c (A)$ of a complex matrix A is defined as the minimal number of additions, subtractions, and multiplications by complex constants of absolute value $ \leqq c$, sufficient to evaluate A at a generic input vector. It is shown that if A is a Fourier transform on the finite group G, then $| L_\infty (A) - L_\infty (A^{ - 1} ) | \leqq | G |$. $L_c (G): = \min \{ L_c (A)|A{\text{ a Fourier transform for }}G\} $ is called the c-linear complexity of the finite group G. It is proved that $L_2 (G) > \frac{1}{4}| G |\log | G |$ for any finite group G, and two infinite classes of non-abelian groups G with $L_2 (G) \leqq 0.6| G |\log | G |$ and $L_2 (G) \leqq 0.8| G |\log | G |$, respectively, are presented. Thus there are non-abelian groups with even faster Fourier transforms than elementary abelian 2-groups (for which $L_2 (G) \leqq | G |\log | G |$)!

Small asymmetric sumsets in elementary abelian 2-groups

For an odd prime $p$, a $p$-transposition group is a group generated by a set of involutions such that the product of any two has order 2 or $p$. We first classify a family of $(G,2)$-geodesic transitive Cayley graphs ${\rm\Gamma}:=\text{Cay}(T,S)$ where $S$ is a set of involutions and $T:\text{Inn}(T)\leq G\leq T:\text{Aut}(T,S)$. In this case, $T$ is either an elementary abelian 2-group or a $p$-transposition group. Then under the further assumption that $G$ acts quasiprimitively on the vertex set of ${\rm\Gamma}$, we prove that: (1) if ${\rm\Gamma}$ is not $(G,2)$-arc transitive, then this quasiprimitive action is the holomorph affine type; (2) if $T$ is a $p$-transposition group and $S$ is a conjugacy class, then $p=3$ and ${\rm\Gamma}$ is $(G,2)$-arc transitive.