Elementary Abelian Research Articles

Lifting elementary Abelian covers of curves

Given a Galois cover of curves f over a field of characteristic p, the lifting problem asks whether there exists a Galois cover over a complete mixed characteristic discrete valuation ring whose reduction is f. In this paper, we consider the case where the Galois groups are elementary abelian p-groups. We prove a combinatorial criterion for lifting an elementary abelian p-cover, dependent on the branch loci of lifts of its p-cyclic subcovers. We also study how branch points of a lift coalesce on the special fiber. Finally, for p=2, we analyze lifts for several families of (Z/2)3-covers of various conductor types, both with equidistant branch locus geometry and non-equidistant branch locus geometry.

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

A family of \(2\)-groups and an associated family of semisymmetric, locally \(2\)-arc-transitive graphs

A mixed dihedral group is a group \(H\) with two disjoint subgroups \(X\) and \(Y\), each elementary abelian of order \(2^n\), such that \(H\) is generated by \(X\cup Y\), and \(H/H'\cong X\times Y\). In this paper, for each \(n\geq 2\), we construct a mixed dihedral \(2\)-group \(H\) of nilpotency class \(3\) and order \(2^a\) where \(a=(n^3+n^2+4n)/2\), and a corresponding graph \(\Sigma\), which is the clique graph of a Cayley graph of \(H\). We prove that \(\Sigma\) is semisymmetric, that is, \({\mathop{\rm Aut}}(\Sigma)\) acts transitively on the edges but intransitively on the vertices of \(\Sigma\). These graphs are the first known semisymmetric graphs constructed from groups that are not \(2\)-generated (indeed \(H\) requires \(2n\) generators). Additionally, we prove that \(\Sigma\) is locally \(2\)-arc-transitive, and is a normal cover of the `basic' locally \(2\)-arc-transitive graph \({\rm\bf K}_{2^n,2^n}\). As such, the construction of this family of graphs contributes to the investigation of normal covers of prime-power order of basic locally \(2\)-arc-transitive graphs – the `local' analogue of a question posed by C. H. Li.

Elementary Abelian subgroups in 2-groups whose derived subgroups are Klein 4-groups

Let P be a finite 2-group. Let be a poset consisting of elementary abelian subgroups of rank at least 2. If the derived subgroup , we proved the spheres occurring in all have the same dimension.

Genus Field and Extended Genus Field of an Elementary Abelian Extension of Global Fields

In the present work we give the construction of the genus field and the extended genus field of an elementary abelian l-extension of a field of rational functions, where l is a prime number. In the Kummer case, if K is contained in a cyclotomic function field, the construction is given using Leopoldt’s ideas by means of Dirichlet characters. Following the definition of Anglès and Jaulent of extended Hilbert class field, we obtain the extended genus field of an elementary abelian l-extension of a field of rational functions.

Conjugacy classes of maximal cyclic subgroups

Abstract In this paper, we study the number of conjugacy classes of maximal cyclic subgroups of a finite group 𝐺, denoted η ⁢ ( G ) \eta(G) . First we consider the properties of this invariant in relation to direct and semi-direct products, and we characterize the normal subgroups 𝑁 with η ⁢ ( G / N ) = η ⁢ ( G ) \eta(G/N)=\eta(G) . In addition, by applying the classification of finite groups whose nontrivial elements have prime order, we determine the structure of G / ⟨ G − ⟩ G/\langle G^{-}\rangle , where G − G^{-} is the set of elements of 𝐺 generating non-maximal cyclic subgroups of 𝐺. More precisely, we show that G / ⟨ G − ⟩ G/\langle G^{-}\rangle is either trivial, elementary abelian, a Frobenius group or isomorphic to A 5 A_{5} .

Elementary abelian covers of the Wreath graph W (3, 2) and the Foster graph F26A

Arc-transitive and edge-transitive graphs are widely used in computer networks. Therefore, it is very useful to introduce and study the properties of these graphs. A graph can be called -edge-transitive (G-E-T) or -arc-transitive (G-A-T) if acts transitively on its edges or arc set, where respectively. A regular covering projection (C-P) is E-T or A-T if an E-R or A-T subgroup of lifts under In this paper, we first study all -elementary abelian (E-A) regular covers of Wreath graph and then investigate (E-T) regular -covers of the Foster graph

On the Schur multiplier of finite 𝑝-groups of maximal class

Abstract In this article, we prove that the Schur multiplier of a finite 𝑝-group of maximal class of order p n p^{n} ( 4 ≤ n ≤ p + 1 4\leq n\leq p+1 ) is elementary abelian. The case n = p + 1 n=p+1 settles a question raised by Primož Moravec in an earlier article.

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.

Classification of Arc-Transitive Elementary Abelian Covers of the C13 Graph

Let Γ be a graph and G⩽Aut(Γ). A graph Γ can be called G-arc-transitive (GAT) if G acts transitively on its arc set. A regular covering projection p:Γ¯→Γ is arc-transitive (AT) if an AT subgroup of Aut(Γ) lifts under p. In this study, by applying a number of concepts in linear algebra such as invariant subspaces (IVs) of matrix groups (MGs), we discuss regular AT elementary abelian covers (R-AT-EA-covers) of the C13 graph.

Some necessary and sufficient conditions on commuting automorphisms of some finite p-groups

Let G be a finite non-abelian p-group of order greater than p 4, p an odd prime, such that is cyclic and Z(H) is elementary abelian, where We prove that the set of all commuting automorphisms of G forms a subgroup of if and only if is abelian. Also, we find the structure of for a finite 2-group G of almost maximal class with cyclic center Z(G), where denotes the set of all central automorphisms of G.

On AS-configurations, skew-translation generalised quadrangles of even order and a conjecture of Payne

Generalised quadrangles, as a special case of generalised polygons, were introduced by Jacques Tits [14] in 1959. About twenty years later Kantor [7] showed that an elation generalised quadrangle is completely characterized by its so-called 4-gonal family or Kantor family.Motivated by the Ahrens and Szekeres Quadrangle [1], Ghinelli [4] established a variation of the procedure of Kantor [7], by introducing the notion of an AS-configuration of order n. Such a configuration gives rise to a skew-translation generalised quadrangle of order (n,n), and conversely, outlined in J. Bamberg, S.P. Glasby, E. Swartz [2]. The only known groups of even order admitting an AS-configuration are elementary abelian 2-groups. Moreover, Payne [8] conjectured that there are no other examples. Indeed, he found a proof for n=4. Moreover it is also true for n=8, compare J. Bamberg, S.P. Glasby, E. Swartz [2].The purpose of this paper is to proveTheoremA group of even order admitting an AS-configuration of order n is elementary abelian, if n is not a square. In other words, the only skew-translation generalised quadrangles of order (n,n), n≡0(mod2), are translation generalised quadrangles, if n is not a square.K. Thas claims in his preprint [12] that he found among other things a proof of Payne's conjecture. Unfortunately, in the main part of his proof there is a gap not easy to fill, [13].

On 2-maximal subgroups of finite groups

We distinguish 2-maximal and strictly 2-maximal subgroups of a finite group and give examples of soluble and simple groups in which every 2-maximal subgroup is strictly 2-maximal. Let M be a maximal subgroup of a group G. We prove that every maximal subgroup of M is strictly 2-maximal in G if M is normal in G or if G is p-soluble and We describe the structure of a finite group in which all 2-maximal subgroups are Hall subgroups. In particular, it has a Sylow tower and all its Sylow subgroups are elementary abelian.

Counterexamples to “A conjecture on induced subgraphs of Cayley graphs”

Recently, Huang gave a very elegant proof of the Sensitivity Conjecture by proving that hypercube graphs have the following property: every induced subgraph on a set of more than half the vertices has maximum degree at least √d , where d is the valency of the hypercube. This was generalised by Alon and Zheng who proved that every Cayley graph on an elementary abelian 2 -group has the same property. Very recently, Potechin and Tsang proved an analogous results for Cayley graphs on abelian groups. They also conjectured that all Cayley graphs have the analogous property. We disprove this conjecture by constructing various counterexamples, including an infinite family of Cayley graphs of unbounded valency which admit an induced subgraph of maximum valency 1 on a set of more than half the vertices.

A Recursive Construction for Skew Hadamard Difference Sets

A major conjecture on the existence of abelian skew Hadamard difference sets is: if an abelian group $G$ contains a skew Hadamard difference set, then $G$ must be elementary abelian. This conjecture remains open in general. In this paper, we give a recursive construction for skew Hadamard difference sets in abelian (not necessarily elementary abelian) groups. The new construction can be considered as a result on the aforementioned conjecture: if there exists a counterexample to the conjecture, then there exist infinitely many counterexamples to it.

On the lattice of subquandles of a Takasaki quandle

Let A be an abelian group. If for any we define , then is a quandle called the Takasaki quandle and is denoted by T(A). In this article, we focus on the lattice of subquandles of a finite Takasaki quandle T(A), denoted by . Indeed, we characterize the subquandles of T(A) and show that is the lattice of cosets of A if and only if A is an odd abelian group or its 2-Sylow subgroup is a cyclic group. Further, we determine the homotopy type of the lattice of subquandles of T(A) which is the wedge of r spheres of dimension d, where r is the number of prime numbers dividing and d is determined by applying the fundamental theorem of finite abelian groups. Furthermore, we prove that is graded if and only if A is an odd abelian group or its 2-Sylow subgroup is cyclic or its 2-Sylow subgroup is elementary abelian. Moreover, we show that the lattice of subquandles of a finite Takasaki quandle is determined by a unique Takasaki quandle. In other words, we show that the lattices of subquandles of two finite Takasaki quandles T(A) and T(B) are isomorphic if and only if A and B are isomorphic. However, the lattice of subquandles of a quandle which is not a Takasaki quandle can be isomorphic to the lattice of subquandles of a Takasaki quandle. We show this by providing a quandle Q of order 8 which is not a Takasaki quandle, but is isomorphic to

Элементарные абелевы 2-подгруппы в группе автотопизмов полуполевой проективной плоскости

Элементарные абелевы 2-подгруппы в группе автотопизмов полуполевой проективной плоскости

Lattices of Finite Abelian Groups

We study certain lattices constructed from finite abelian groups. We show that such a lattice is eutactic, thereby confirming a conjecture by B\"ottcher, Eisenbarth, Fukshansky, Garcia, Maharaj. Our methods also yield simpler proofs of two known results: First, such a lattice is strongly eutactic if and only if the abelian group has odd order or is elementary abelian. Second, such a lattice has a basis of minimal vectors, except for the cyclic group of order 4.

Doubly transitive dimensional dual hyperovals: universal covers and non-bilinear examples

AbstractBy [4] a doubly transitive, non-solvable dimensional dual hyperovalDis isomorphic either to the Mathieu dual hyperoval or to a quotient of a Huybrechts dual hyperoval. In order to determine all doubly transitive dimensional dual hyperovals, it remains to classify the solvable ones, and this paper is a contribution to this problem. A doubly transitive, solvable dimensional dual hyperovalDof ranknis defined over 𝔽2and has an automorphism of the formES, whereEis elementary abelian of order 2nandS≤ Γ L(1, 2n); see Yoshiara [12]. The known examplesDare bilinear. In [1] thebilinear, doubly transitive, solvable dimensional dual hyperovalsDof ranknwith GL(1, 2n) ≤Sare classified. Here we present two new classes ofnon-bilinear, doubly transitive dimensional dual hyperovals. We also consider universal covers of doubly transitive dimensional dual hyperovals, since they are again doubly transitive dimensional dual hyperovals by [2, Cor. 1.3]. We determine the universal covers of the presently known doubly transitive dimensional dual hyperovals.

Some properties on mathrm{IA_Z}-automorphisms of groups

Let G be a group and mathrm{IA}(G) denote the group of all automorphisms of G, which induce identity map on the abelianized group G_{ab}=G/G'. Also the group of those mathrm{IA}-automorphisms which fix the centre element-wise is denoted by mathrm{IA_Z}(G). In the present article, among other results and under some condition we prove that the derived subgroups of finite p-groups, for which mathrm{IA_Z}-automorphisms are the same as central automorphisms, are either cyclic or elementary abelian.