Representation-graded Bredon homology of elementary abelian 2-groups

Nonassociative division algebras play a significant role in Physics and Communications. The finite nonassociative division algebras have a vast range of applications on coding theory, combinatorics and graph theory. This paper deals with a class of finite structures known as division algebras. For a long time division algebras have been studied from a geometric point of view, since they coordinatize certain types of projective planes as an important part of finite geometric incidence. But recent results relating division algebras and coding theory (and also the study of Generalized Galois Rings) have stimulated the study of these rings from a strictly algebraic point of view. This paper follows the second path. Let <i>A</i> be a unital division algebra of order of <i>q<sup>4</sup></i>, q is an odd prime power greater than 3. We assume that <i>A</i> admits an elementary abelian automorphism group <i>E</i> acting freely on <i>A, i.e A≌𝔽<sub>q</sub>[E]</i>. The purpose of this paper is to classify this class of division algebras. In addition, we compute a bound for <i>q</i> and deduce relations among certain structure constants for the quartics associated with <i>A</i>. These relations determine <i>A</i> completely. To achieve these objectives an algebraic geometric approach which is mainly based on the prominent results namely Hasse-Weil theorem and Chevalley-Wraring theorem and the work of Menichetti on n-dimensional algebras over fields of cyclic extensions of degree <i>n</i>.

We study two kinds of generalizations of symmetric block designs to higher dimensions, the so-called C -cubes and P -cubes. For small parameters, all examples up to equivalence are determined by computer calculations. Known properties of automorphisms of symmetric designs are extended to autotopies of P -cubes, while counterexamples are found for C -cubes. An algorithm for the classification of P -cubes with prescribed autotopy groups is developed and used to construct more examples. A bound on the dimension of difference sets for P -cubes is proved and shown to be tight in elementary abelian groups. The construction is generalized to arbitrary groups by introducing regular sets of (anti)automorphisms.

In an earlier work, finite groups whose power graphs are minimally edge connected have been classified. In this article, first we obtain a necessary and sufficient condition for an arbitrary graph to be minimally edge connected. Consequently, we characterize finite groups whose enhanced power graphs and order superpower graphs, respectively, are minimally edge connected. Moreover, for a finite non-cyclic group G, we prove that G is an elementary abelian 2-group if and only if its enhanced power graph is minimally connected. Also, we show that G is a finite p-group if and only if its order superpower graph is minimally connected. Finally, we characterize all the finite nilpotent groups such that the minimum degree and the vertex connectivity of their order superpower graphs are equal.

Abstract This paper is concerned with images of cosets in the direct product of groups by bijective mappings from factors to groups.We prove necessary and sufficient conditions on bijective mappings for existence a coset in the direct product of two groups whose image is a coset. Under some constraints on bijective mappings, we describe the cosets in the direct product of groups, whose images by bijective mappings from factors to groups are cosets, We described the cosets in the direct product of elementary abelian 2-groups whose images by the inversion permutation of nonzero elements of a finite field on factors are cosets. We also describe the similar cosets for the permutation used an an s-box of Kuznyechik algorithm, Under some constraints on bijective mappings, we describe the automorphisms of the direct product groups which commute with bijective mappings from factors to groups.

Transfer systems for rank two elementary abelian groups: characteristic functions and matchstick games

Let $G$ be a finite group and $\mathrm{Irr}(G)$ be the set of irreducible characters of $G$. The codegree of an irreducible character $\chi$ of the group $G$ is defined as $\mathrm{cod}(\chi)=|G:\mathrm{ker}(\chi)|/\chi(1)$. In this paper, we consider the finite group $G$ whose all irreducible character codegrees are consecutive integers $1,2,3,...,k-1,k$. We prove that $k\leq 3$ and $G$ is an elementary abelian group or a Frobenius group.

Let G be a elementary abelian 2-group and X be a manifold with a locally standard action of G. We provide a criterion to determine the syzygy order of the G-equivariant cohomology of X with coefficients over a field of characteristic two using a complex associated to the cohomology of the face filtration of the manifold with corners X{G. This result is the real version of the quotient criterion for locally standard torus actions developed in [16] .

We study Abelian groups whose endomorphism rings are V-rings. Let [Formula: see text] be a non-reduced Abelian group, We prove that [Formula: see text] is a V-ring on either side if and only if [Formula: see text] where [Formula: see text] is a tame elementary Abelian group. We observe that a reduced group whose endomorphism is a V-ring, is an sp-group. Recognizing that [Formula: see text] is also an sp-group of [Formula: see text], we show that [Formula: see text] is a V-ring if and only if [Formula: see text] is a V-ring.

Let ℤ n be the ring of integers modulo n. Let Ct , Em , and F r , s respectively denote the cyclic group of order t, the elementary abelian 2-group of order 2 m , and the abelian group of exponent 4 with order 2 r 4 s . In this article, we find the structure and generators of the unit group V ( ℤ n C 2 ) . We also solve the normal complement problem in V ( ℤ n C 2 ) . Additionally, we provide a normal complement of Em in V ( ℤ 2 n E m ) . At the end, we determine the structure of V ( ℤ p n F r , s ) for an odd prime p and establish that F r , s does not have a normal complement in V ( ℤ p n F r , s ) .

On the zero-sum subsequences of modular restricted lengths

Let Cn be a cyclic group of order n. We investigate K 2 of integral group rings via the Mayer-Vietoris sequence, and give a decomposition of the 2-primary torsion subgroup of K 2 ( Z [ C p × ( C 2 ) n ] ) for any prime p ≡ 3 , 5 , 7 ( mod 8 ) , in particular, K 2 ( Z [ C 3 × ( C 2 ) n ] ) is proven to be a finite abelian 2-group. As an application, we prove K 2 ( Z [ C 3 × ( C 2 ) 2 ] ) is an elementary abelian 2-group of rank at least 14, at most 16.

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)} $.

Let F be a non-cyclic free group of rank n. Consider the quotient F / [ F ″ , F ] , the free centre-by-metabelian group of rank n. In 1973, C. K. Gupta proved by purely group theoretic means that it contains an elementary abelian 2-group of rank ( 4 n ) in its centre for n ≥ 4 , and exhibited an explicit generating set for this torsion subgroup. In this paper, using homological methods, we provide an alternative explicit generating set for it, and identify this torsion subgroup as the isolator of an explicitly given subgroup in the quotient F ″ / [ F ″ , F ′ ] .

This article is concerned with the classification of Schur covering groups of the elementary abelian group of order 2 n , up to isomorphism. We consider those covering groups possessing a generating set of n elements having only two distinct squares. We show that such groups may be represented by 2-vertex-colored and 2-edge-colored graphs of order n. We show that in most cases, the isomorphism type of the group is determined by that of the 2-colored graph, and we analyze the exceptions.

Given a finite group R, we let Sub(R) denote the collection of all subgroups of R. We show that |Sub(R)|<c⋅|R|log2⁡(|R|)4, where c<7.372 is an explicit absolute constant. This result is asymptotically best possible. Indeed, as |R| tends to infinity and R is an elementary abelian 2-group, the ratio|Sub(R)||R|log2⁡(|R|)4 tends to c.

Amorphic association schemes from bent partitions

The Galois embedding problem of an extension with elementary Abelian 2-group into an extension with the Galois group isomorphic to the group of unitriangular matrices over the 2-element field is considered. It is proved that the solvability of the maximal accompanying problem with central kernel of period 2 is sufficient for the solvability of the original problem.

Zigzag polynomials, Artin's conjecture and trinomials

Let $G$ be a fct Lie group and let $U$ and $V$ be finite-dimensional real $G$-modules with $V^G=0$. A theorem of Marzantowicz, de Mattos and dos Santos estimates the covering dimension of the zero-set of a $G$-map from the unit sphere in $U$ to $V$ when $G$ is an elementary abelian $p$-group for some prime $p$ or a torus. In this note, the classical Borsuk-Ulam theorem will be used to give a refinement of their result estimating the dimension of that part of the zero-set on which an elementary abelian $p$-group $G$ acts freely or a torus $G$ acts with finite isotropy groups. The methods also provide an easy answer to a question raised in \cite{DM}.