Small asymmetric sumsets in elementary abelian 2-groups

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.

We show that, for a positive integer r, every minimal 1-saturating set in ${PG}(r-1,2)$ of size at least $\frac{11}{36}\,2^r+3$ either is a complete cap or can be obtained from a complete cap S by fixing some $s\in S$ and replacing every point $s'\in S\setminus\{s\}$ by the third point on the line through s and $s'$. Since, conversely, every set obtained in this way is a minimal 1-saturating set and the structure of large sum-free sets in an elementary abelian 2-group is known, this provides a complete description of large minimal 1-saturating sets. An algebraic restatement is as follows. Suppose that G is an elementary abelian 2-group and a subset $A\subseteq G\setminus\{0\}$ satisfies $A\cup2A=G$ and is minimal subject to this condition. If $|A|\ge\frac{11}{36}\,|G|+3$, then either A is a maximal sum-free set or there are a maximal sum-free set $S\subseteq G$ and an element $s\in S$ such that $A=\{s\}\cup\bigl(s+(S\setminus\{s\})\bigr)$. Our approach is based on characterizing those large sets A in elementary abelian 2-groups such that, for every proper subset B of A, the sumset $2B$ is a proper subset of $2A$.

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.

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

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 $L_2 (G) \leqq | G |\log | G |$)!

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

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.

SynopsisLet Γ be a graph with n points, and let G be the group of automorphisms of Γ. An orbit of G on which G acts as an elementary abelian 2-group is said to be exceptional. It is shown that the number of simple eigenvalues of Γ is at most (5n+4t)/9, where t is the number of points of Γ lying in exceptional orbits of G.

A power automorphism θ of a group G is said to be pre-fixed-point-free if CG(θ) is an elementary abelian 2-group. G is called an E-group if G has a pre-fixed-point-free power automorphism. In this paper, finite E-groups, together with all their pre-fixed-point-free power automorphisms, are completely determined. Moreover, a characteristic of finite abelian groups is given, which explains some known facts concerning power automorphisms.

An Elementary Abelian Group of Rank 4 Is a CI-Group

Elementary abelian groups can be thought of as additive groups of finite fields. As such, all of the tools of field theory are available to us in the study of orthomorphism graphs of these groups. In particular, any function from a finite field to itself, and thus any orthomorphism of the additive group of the field, can be realized as a polynomial function. Several interesting classes of orthomorphisms will be described as sets of orthomorphism polynomials by placing restrictions on the polynomials. Classes of normalized orthomorphisms of additive groups of finite fields will be defined and studied using multiplication. The simplest such class is the class of linear orthomorphisms: these are of the form x↦ax. It is clear that such a mapping is an orthomorphism if and only if a≠0, 1. In this chapter we will also study quadratic orthomorphisms: these map x to ax if x is a square and to bx if x is a nonsquare.