Elementary Abelian Group Research Articles

The Quotient Criterion for Syzygies in Equivariant Cohomology for Elementary Abelian 2-Group Actions

The Quotient Criterion for Syzygies in Equivariant Cohomology for Elementary Abelian 2-Group Actions

Abelian groups whose endomorphism rings are V-rings

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.

The unit group of the group ring over ℤn

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 G be an additive finite abelian group and let ℓ be a positive integer. Denote by discℓ(G) the smallest positive integer t such that every sequence S over G of length |S|≥t has a nonempty zero-sum subsequence T with length |T|≠ℓ. Let disc(G) denote the smallest positive integer t such that every sequence S over G of length |S|≥t has two nonempty zero-sum subsequences of distinct lengths. Gao et al. [8] proved that disc(G)=max⁡{discℓ(G):ℓ∈N0}. In this paper, we continue to investigate this invariant discℓ(G) by introducing a new invariant Ek,u(G). Let k and u be positive integers with k≥2 and u∈[1,k]. Denote by Ek,u(G) the smallest positive integer t such that every sequence S over G of length |S|≥t has a nonempty zero-sum subsequence T with length |T|≢u (mod k). In addition to consider the exact value and inverse problem of discℓ(G), we also study the relationship between Ek,u(G) and disc(G), discℓ(G) for various types of abelian groups. In particular, we determine the exact value of Ek,u(G) for G=Cn with u∈[1,k−1] and that of Ek,1(G) for elementary abelian 2-groups G=C2r, respectively.

A note on one decomposition of the 2-primary part of K 2(ℤ[C p × (C 2) n

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 for generalized Boolean functions

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)} $.

An alternative description of the torsion subgroup of the free centre-by-metabelian group of rank at least four in terms of generators

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 ′ ] .

2-Uniform covering groups of elementary abelian 2-groups

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.

On the maximum number of subgroups of a finite group

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

Bent partitions are partitions of an elementary abelian group, which have similarities to partitions from spreads. In fact, a spread partition is a special case of a bent partition. In particular, bent partitions give rise to a large number of (vectorial) bent functions. Examples of bent partitions, which generalize the Desarguesian spread, have been presented by Anbar, Meidl and Pirsic, 2021, 2022. Bent partitions, which generalize some other classes of (pre)semifield spreads, have been presented by Anbar, Kalaycı, Meidl 2023. In this article, it is shown that these bent partitions induce (pk+1)-class amorphic associations schemes on Fpm×Fpm, where k is a divisor of m with special properties. This implies a construction of amorphic association schemes from some classes of (pre)semifields.

About One Galois Embedding Problem

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 Fp denote the finite field of prime order p. We introduce a new natural family of polynomials in Fp[X]. These polynomials first arose from considering faithful representations of the elementary abelian group of order p2 in characteristic p. The polynomials exhibit a number of interesting special properties. For example they satisfy a three term recursion, are closely related to zigzag zero-one sequences and form strong divisibility sequences. These polynomials are shown to be closely connected to the order of appearance of prime numbers in the Fibonacci sequence, Artin's Primitive Root Conjecture, and the factorization of trinomials over finite fields.

Borsuk-Ulam theorems for elementary abelian 2-groups

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}.

CI-property of [formula omitted] and [formula omitted] for digraphs

We prove that the direct product of two coprime order elementary abelian groups of rank two, as well as the direct product of a cyclic group of prime order and a cyclic group of square-free order are DCI-groups. The latter is a generalization of Muzychuk's result on cyclic groups (J. Combin. Theory Ser. A, 1995).

Doubly transitive, bilinear dimensional dual hyperovals: The Conclusion

In Yoshiara (2009) [17] determined possible groups which can act as a doubly transitive group of automorphisms on a dimensional dual hyperoval. The non-solvable, doubly transitive dual hyperoval were classified in Dempwolff (2018). By Yoshiara (2009) solvable, doubly transitive dual dimensional hyperovals are dual hyperovals defined over F2. If such a hyperoval has rank n, the doubly transitive group has the form G=E⋅H (semidirect product), E is a normal, elementary abelian group of order 2n in G and H≤ΓL(1,2n) acts transitively by conjugation on the non-trivial elements of E. The solvable, doubly transitive, dual hyperovals have been classified by Yoshiara (2008) and the author (Dempwolff 2015) provided the dual hyperoval is bilinear and GL(1,2n)≤H. In particular the doubly transitive, bilinear dual hyperovals are known if (n,2n−1)=1. If (n,2n−1)≠1 the group ΓL(1,2n) contains subgroups H, such that GL(1,2n)⁄≤H and H acts transitively by conjugation on the non-trivial elements of E. Our paper addresses this case and closes the investigation of doubly transitive, bilinear dual hyperovals.

Elementary abelian group actions on a product of spaces of cohomology type (a, b)

Let Xn be a finite CW complex with cohomology type (a, b), characterized by an integer n &gt; 1 [20]. In this paper, we show that if G = (Z2)q acts freely on the product Y = Qmi=1 Xin, where Xin are finite CW complexes with cohomology type (a, b), a and b are even for every i, then q ? m. Moreover, for n even and a = b = 0, we prove that G = (Z2)q (q ? m) is the only finite group which can act freely on Y. These are generalizations of the results which says that the rank of a group acting freely on a space with cohomology type (a, b) where a and b are even, is one and for n even, G = Z2 is the only finite group which acts freely on spaces of cohomology type (0, 0) [17].

On the average order of a finite group

Let o(G) be the average order of a finite group G. We show that if o(G)<c, where c∈{136,114}, then G is an elementary abelian 2-group or a solvable group, respectively. Also, we prove that the set containing the average orders of all finite groups is not dense in [a,∞), for all a∈[0,136]. We also outline some results related to the integer values of the average order. Since group element orders is a popular research topic, we pose some open problems concerning the average order of a finite group throughout the paper.

Formula omitted]-property of some groups

In this paper, firstly, we determine the local 2-GCI-property and 2−GCI-property of the cyclic group. Then, for the dihedral group D2n, we prove that it has local GCI-property if and only if n is an odd prime or 9. Further, the dihedral group D2n cannot have GCI-property. Moreover, we discuss the GCI-property of the elementary abelian group, the dicyclic group and the semi-dihedral group.

ON AUTOMORPHISM GROUPS OF ENDOMORPHISM SEMIGROUPS OF FINITE ELEMENTARY ABELIAN GROUPS

In this article, we explore the automorphisms of endomorphism semigroups and automorphism groups of the finite elementary Abelian groups. In particular, we prove that $\mathrm{Aut}(\mathrm{End}(\Z_p\oplus\Z_p\oplus\cdots\oplus\Z_p))$ can be canonically embedded into $\mathrm{Aut}(\mathrm{Aut}(\Z_p\oplus\Z_p\oplus\cdots\oplus\Z_p))$ using an elementary approach based on matrix operations. We also show that all automorphisms of $\mathrm{End}(\Z_p\oplus\Z_p\oplus\cdots\oplus\Z_p)$ are inner.

Parallelisms of PG(3,4) invariant under an elementary abelian group of order 4

Parallelisms of PG(3,4) invariant under an elementary abelian group of order 4