Elementary Abelian P-group 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.

Modular invariants of a vector and a covector for some elementary abelian p-groups

Let F p be the prime field of order p > 0 and G be an elementary abelian p-group. For some n-dimensional cohyperplane G-representations V over F p , we show that F p [ V ⊕ V * ] G , the invariant ring of a vector and a covector is a complete intersection by exhibiting an explicit generating set (in fact, a SAGBI basis) and exposing all relations among the generators.

Skew-morphisms of elementary abelian 𝑝-groups

Abstract A skew-morphism of a finite group 𝐺 is a permutation 𝜎 on 𝐺 fixing the identity element, and for which there exists an integer-valued function 𝜋 on 𝐺 such that σ ⁢ ( x ⁢ y ) = σ ⁢ ( x ) ⁢ σ π ⁢ ( x ) ⁢ ( y ) \sigma(xy)=\sigma(x)\sigma^{\pi(x)}(y) for all x , y ∈ G x,y\in G . It is known that, for a given skew-morphism 𝜎 of 𝐺, the product of the left regular representation of 𝐺 with ⟨ σ ⟩ \langle\sigma\rangle forms a permutation group on 𝐺, called a skew-product group of 𝐺. In this paper, we study the skew-product groups 𝑋 of elementary abelian 𝑝-groups Z p n \mathbb{Z}_{p}^{n} . We prove that 𝑋 has a normal Sylow 𝑝-subgroup and determine the structure of 𝑋. In particular, we prove that Z p n ⊲ X \mathbb{Z}_{p}^{n}\lhd X if p = 2 p=2 and either Z p n ⊲ X \mathbb{Z}_{p}^{n}\lhd X or ( Z p n ) X ≅ Z p n − 1 (\mathbb{Z}_{p}^{n})_{X}\cong\mathbb{Z}_{p}^{n-1} if 𝑝 is an odd prime. As an application, for n ≤ 3 n\leq 3 , we prove that 𝑋 is isomorphic to a subgroup of the affine group AGL ⁢ ( n , p ) \mathrm{AGL}(n,p) and enumerate the number of skew-morphisms of Z p n \mathbb{Z}_{p}^{n} .

Krull dimension of the negligible quotient in mod p cohomology of a finite group

For a finite group G, a G-module M, and a field F, an element u∈Hd(G,M) is negligible over F if for each field extension L/F and every continuous group homomorphism from Gal(Lsep/L) to G, u belongs to the kernel of the induced homomorphism Hd(G,M)→Hd(L,M). For p a prime and a trivial G-action on the coefficients, the negligible elements in the cohomology ring H⁎(G,Z/pZ) form an ideal. We compute the generators of the negligible ideal in the mod p cohomology of elementary abelian p-groups. We further show that when p is odd or p=2 and either |G| is odd or F is not formally real, the Krull dimension of the quotient of mod p cohomology by the negligible ideal is 0. However, when p=2, |G| is even, and F is formally real, the Krull dimension of the quotient of mod 2 cohomology of a finite 2-group by the negligible ideal is 1.

On the occurrence of elementary abelian p-groups as the Schur multiplier of non-abelian p-groups

We prove that every elementary abelian p-group, for odd primes p, occurs as the Schur multiplier of some non-abelian finite p-group.

Regular Cayley maps of elementary abelian p-groups: Classification and enumeration

Recently, regular Cayley maps of cyclic groups and dihedral groups have been classified in [7] and [20], respectively. A nature question is to classify regular Cayley maps of elementary abelian p-groups Zpn. In this paper, a complete classification of regular Cayley maps of Zpn is given and moreover, the number of these maps and their genera are enumerated.

Small modules with interesting rank varieties

This paper focuses on the rank varieties for modules over a group algebra FE where E is an elementary abelian p-group and p is the characteristic of an algebraically closed field F. In the first part, we give a sufficient condition for a Green vertex of an indecomposable module to contain an elementary abelian p-group E in terms of the rank variety of the module restricted to E. In the second part, given a homogeneous algebraic variety V, we explore the problem on finding a small module with rank variety V. In particular, we examine the simple module D(kp−p+1,1p−1) for the symmetric group Skp.

Determination of compact Lie groups with the Borsuk-Ulam property

A compact Lie group G is said to have the Borsuk-Ulam property if the Borsuk-Ulam theorem holds for G-maps between representation spheres. It is well-known that an elementary abelian p-group Cpn (p any prime) and an n-torus Tn have the Borsuk-Ulam property. In this paper, we shall discuss the classical question of which compact Lie groups have the Borsuk-Ulam property and in particular we shall show that every extension group of an n-torus by a cyclic group of prime order does not have the Borsuk-Ulam property. This leads us that the only compact Lie groups with the Borsuk-Ulam property are Cpn and Tn, which is a final answer to the question.

Dualizing spheres for compact p-adic analytic groups and duality in chromatic homotopy

The primary goal of this paper is to study Spanier–Whitehead duality in the K(n)-local category. One of the key players in the K(n)-local category is the Lubin–Tate spectrum \(E_n\), whose homotopy groups classify deformations of a formal group law of height n, in the implicit characteristic p. It is known that \(E_n\) is self-dual up to a shift; however, that does not fully take into account the action of the automorphism group \({\mathbb {G}}_n\) of the formal group in question. In this paper we find that the \({\mathbb {G}}_n\)-equivariant dual of \(E_n\) is in fact \(E_n\) twisted by a sphere with a non-trivial (when \(n>1\)) action by \({\mathbb {G}}_n\). This sphere is a dualizing module for the group \({\mathbb {G}}_n\), and we construct and study such an object \(I_{{\mathcal {G}}}\) for any compact p-adic analytic group \({\mathcal {G}}\). If we restrict the action of \({\mathcal {G}}\) on \(I_{{\mathcal {G}}}\) to certain type of small subgroups, we identify \(I_{{\mathcal {G}}}\) with a specific representation sphere coming from the Lie algebra of \({\mathcal {G}}\). This is done by a classification of p-complete sphere spectra with an action by an elementary abelian p-group in terms of characteristic classes, and then a specific comparison of the characteristic classes in question. The setup makes the theory quite accessible for computations, as we demonstrate in the later sections of this paper, determining the K(n)-local Spanier–Whitehead duals of \(E_n^{hH}\) for select choices of p and n and finite subgroups H of \({\mathbb {G}}_n\).

On a maximal subgroup of the orthogonal group O⁺₈(3)

The orthogonal simple group 0 (3) has three conjugacy classes of maximal subgroups of the form 36:L4(3). These groups are all isomorphic to each other and each group has order 4421589120 with index 1120 in 0 (3). In this paper, we will compute the ordinary carácter table of one of these classes of maximal subgroups using the technique of Fischer-Clifford matrices. This technique is very efficient to compute the ordinary character table of an extension group Ḡ = N.G and especially where the normal subgroup N of Ḡ is an elementary abelian p-group. The said technique reduces the computation of the ordinary character table of Ḡ to find a handful of so-called Fischer-Clifford matrices of Ḡ and the ordinary or projective character tables of the inertia factor groups of the action of Ḡ on N.

Groups whose Chermak–Delgado lattice is a subgroup lattice of an elementary abelian p-group

The Chermak–Delgado lattice of a finite group G is a self-dual sublattice of the subgroup lattice of G. In this paper, we focus on finite groups whose Chermak–Delgado lattice is a subgroup lattice of an elementary abelian p-group. We prove that such groups are nilpotent of class 2. We also prove that, for any elementary abelian p-group E, there exists a finite group G such that the Chermak–Delgado lattice of G is a subgroup lattice of E.

Schurity of association schemes whose thin residues are elementary abelian p-groups of rank 2

The schurity of association schemes has been studied in many papers. One of the major topics is to investigate the schurity of those association schemes whose thin residues are thin. A difficult case is that the thin residue is an elementary abelian p-group of rank 2. A class of these association schemes has played an important role in the study of p-schemes of order $$p^3$$ . In this paper, we study the automorphism groups and schurity problem of this class of association schemes. In particular, we will establish very simple sufficient (and necessary) conditions for these association schemes to be schurian. As an application, we obtain two infinite families of schurian association schemes.

Computing the number of symmetric colorings of elementary Abelian groups

Given a finite group G and a positive integer r, an r-coloring of G is any mapping χ:G→{1,…,r}. Colorings χ and φ are equivalent if there exists g∈G such that χ(xg-1)=φ(x) for all x∈G. A coloring χ is symmetric if there exists g∈G such that χ(gx-1g)=χ(x) for every x∈G. We compute the number of symmetric r-colorings and the number of equivalence classes of symmetric r-colorings of an elementary Abelian p-group.

Modular invariants of finite gluing groups

We use the gluing construction introduced by Jia Huang to explore the rings of invariants for a range of modular representations. We construct generating sets for the rings of invariants of the maximal parabolic subgroups of a finite symplectic group and their common Sylow p-subgroup. We also investigate the invariants of singular finite classical groups. We introduce parabolic gluing and use this construction to compute the invariant field of fractions for a range of representations. We use thin gluing to construct faithful representations of semidirect products and to determine the minimum dimension of a faithful representation of the semidirect product of a cyclic p-group acting on an elementary abelian p-group.

Free [formula omitted]-complexes and p-DG modules

We reformulate the problem of bounding the total rank of the homology of perfect chain complexes over the group ring Fp[G] of an elementary abelian p-group G in terms of commutative algebra. This extends results of Carlsson for p=2 to all primes. As an intermediate step, we construct an embedding of the derived category of perfect chain complexes over Fp[G] into the derived category of p-DG modules over a polynomial ring.

Additivity of affine designs

We show that any affine block design $$\mathcal{D}=(\mathcal{P},\mathcal{B})$$ is a subset of a suitable commutative group $${\mathfrak {G}}_\mathcal{D},$$ with the property that a k-subset of $$\mathcal{P}$$ is a block of $$\mathcal{D}$$ if and only if its k elements sum up to zero. As a consequence, the group of automorphisms of any affine design $$\mathcal{D}$$ is the group of automorphisms of $${\mathfrak {G}}_\mathcal{D}$$ that leave $$\mathcal P$$ invariant. Whenever k is a prime p, $${\mathfrak {G}}_\mathcal{D}$$ is an elementary abelian p-group.

Symplectic reflection algebras in positive characteristic as Ore extensions

We investigate PBW deformations of where G is the cyclic group of order p and the ground field k has characteristic p. The algebras are a version of symplectic reflection algebras that only exist in positive characteristic. They also admit a presentation as Ore extensions over and the combinatorics of the derivation used in this presentation is related to André and Eulerian polynomials. We find the center of and classify the simple modules. We also study a version of in which G is replaced by an elementary abelian p-group Gr.

Indecomposable Jordan types of Loewy length 2

Let k be an algebraically closed field, char(k)=p≥2 and Er be a elementary abelian p-group of rank r≥2. Let (c,d)∈N2. We show that there exists an indecomposable module of constant Jordan type [1]c[2]d and Loewy length 2 if and only if qΓr(d,d+c)≤1 and c≥r−1, where qΓr(x,y):=x2+y2−rxy denotes the Tits form of the generalized Kronecker quiver Γr.Since p>2 and constant Jordan type [1]c[2]d imply Loewy length ≤2, we get in this case the full classification of Jordan types [1]c[2]d that arise from indecomposable modules.

Two-valenced association schemes and the Desargues theorem

The main goal of the paper is to establish a sufficient condition for a two-valenced association scheme to be schurian and separable. To this end, an analog of the Desargues theorem is introduced for a noncommutative geometry defined by the scheme in question. It turns out that if the geometry has enough many Desarguesian configurations, then under a technical condition, the scheme is schurian and separable. This result enables us to give short proofs for known statements on the schurity and separability of quasi-thin and pseudocyclic schemes. Moreover, by the same technique, we prove a new result: given a prime p, any {1,p}-scheme with thin residue isomorphic to an elementary abelian p-group of rank greater than two, is schurian and separable.

Degree three unramified cohomology groups and Noether's problem for groups of order 243

Let k be a field and G be a finite group acting on the rational function field k(xg:g∈G) by k-automorphisms defined as h(xg)=xhg for any g,h∈G. We denote the fixed field k(xg:g∈G)G by k(G). Noether's problem asks whether k(G) is rational (= purely transcendental) over k. It is well-known that if C(G) is stably rational over C, then all the unramified cohomology groups Hnri(C(G),Q/Z)=0 for i≥2. Hoshi, Kang and Kunyavskii [32] showed that, for a p-group of order p5 (p: an odd prime number), Hnr2(C(G),Q/Z)≠0 if and only if G belongs to the isoclinism family Φ10. When p is an odd prime number, Peyre [55] and Hoshi, Kang and Yamasaki [33] exhibit some p-groups G which are of the form of a central extension of certain elementary abelian p-group by another one with Hnr2(C(G),Q/Z)=0 and Hnr3(C(G),Q/Z)≠0. However, it is difficult to tell whether Hnr3(C(G),Q/Z) is non-trivial if G is an arbitrary finite group. In this paper, we are able to determine Hnr3(C(G),Q/Z) where G is any group of order p5 with p=3,5,7. Theorem 1. Let G be a group of order 35. Then Hnr3(C(G),Q/Z)≠0 if and only if G belongs to the isoclinism family Φ7. Theorem 2. If G is a group of order 35, then the fixed field C(G) is rational if and only if G does not belong to the isoclinism families Φ7 and Φ10. Theorem 3. Let G be a group of order 55 or 75. Then Hnr3(C(G),Q/Z)≠0 if and only if G belongs to the isoclinism families Φ6, Φ7 or Φ10. Theorem 4. If G is the alternating group An, the Mathieu group M11, M12, the Janko group J1 or the group PSL2(Fq), SL2(Fq), PGL2(Fq) (where q is a prime power), then Hnrd(C(G),Q/Z)=0 for any d≥2. Besides the degree three unramified cohomology groups, we compute also the stable cohomology groups.