Elementary Abelian Group Research Articles

Examples of finite free complexes of small rank and small homology

This work concerns finite free complexes over commutative noetherian rings, in particular over group algebras of elementary abelian groups. The main contribution is the construction of complexes such that the total rank of their underlying free modules, or the total length of their homology, is less than predicted by various conjectures in the theory of transformation groups and in local algebra.

The chromatic number of the square of the $8$-cube

A cube-like graph is a Cayley graph for the elementary abelian group of order $2^n$. In studies of the chromatic number of cube-like graphs, the $k$th power of the $n$-dimensional hypercube, $Q_n^k$, is frequently considered. This coloring problem can be considered in the framework of coding theory, as the graph $Q_n^k$ can be constructed with one vertex for each binary word of length $n$ and edges between vertices exactly when the Hamming distance between the corresponding words is at most $k$. Consequently, a proper coloring of $Q_n^k$ corresponds to a partition of the $n$-dimensional binary Hamming space into codes with minimum distance at least $k+1$. The smallest open case, the chromatic number of $Q_8^2$, is here settled by finding a 13-coloring. Such 13-colorings with specific symmetries are further classified.

The geometry of the Artin-Schreier-Mumford curves over an algebraically closed field

For a power $q$ of a prime $p$, the Artin-Schreier-Mumford curve $ASM(q)$ of genus $g=(q-1)^2$ is the nonsingular model $\mathcal{X}$ of the irreducible plane curve with affine equation $(X^q+X)(Y^q+Y)=c,\, c\neq 0,$ defined over a field $\mathbb{K}$ of characteristic $p$. The Artin-Schreier-Mumford curves are known from the study of algebraic curves defined over a non-Archimedean valuated field since for $|c|<1$ they are curves with a large solvable automorphism group of order $2(q-1)q^2 =2\sqrt{g}(\sqrt{g}+1)^2$, far away from the Hurwitz bound $84(g-1)$ valid in zero characteristic. In this paper we deal with the case where $\mathbb{K}$ is an algebraically closed field of characteristic $p$. We prove that the group $Aut(\mathcal{X})$ of all automorphisms of $\mathcal{X}$ fixing $\mathbb{K}$ elementwise has order $2q^2(q-1)$ and it is the semidirect product $Q\rtimes D_{q-1}$ where $Q$ is an elementary abelian group of order $q^2$ and $D_{q-1}$ is a dihedral group of order $2(q-1)$. For the special case $q=p$, this result was proven by Valentini and Madan. Furthermore, we show that $ASM(q)$ has a nonsingular model $\mathcal{Y}$ in the three-dimensional projective space $PG(3,\mathbb{K})$ which is neither classical nor Frobenius classical over the finite field $\mathbb{F}_{q^2}$.

The nonabelian tensor square of a bieberbach group with elementary Abelian 2-group point group

A Bieberbach group is a torsion free crystallographic. In this paper, one Bieberbach group with elementary abelian 2-group point group of the lowest dimension three is considered and its group presentation can be shown to be consistent polycyclic presentation. The main objective of this paper is to compute the nonabelian tensor square of one Bieberbach group with elementary abelian 2-group point group of dimension three by using the computational method of the nonabelian tensor square for polycyclic groups. The finding of the computation showed that the nonabelian tensor square of the group is abelian and the formula of the nonabelian tensor square of the Bieberbach group with elementary abelian 2-group of dimension 3, S 1 (3) , can be extended in constructing the generalization of the formula of the nonabelian tensor square of the group up to dimension n . Keywords : Bieberbach group; polycyclic groups; nonabelian tensor square.

The Grothendieck--Teichm\"{u}ller group of~PSL(2,q)

Abstract In this paper, we show that the Grothendieck–Teichmüller group of PSL ⁡ ( 2 , q ) {\operatorname{PSL}(2,q)} , or more precisely the group 𝒢 ⁢ 𝒯 1 ⁢ ( PSL ⁡ ( 2 , q ) ) {\mathcal{G\kern-0.569055ptT}_{\kern-1.707165pt1}(\operatorname{PSL}(2,q))} as previously defined by the author, is the product of an elementary abelian 2-group and several copies of the dihedral group of order 8. Moreover, when q is even, we show that it is trivial. We explain how it follows that the moduli field of any “dessin d’enfant” whose monodromy group is PSL ⁡ ( 2 , q ) {\operatorname{PSL}(2,q)} has derived length ≤ 3 {\leq 3} . This paper can serve as an introduction to the general results on the Grothendieck–Teichmüller group of finite groups obtained by the author.

On a family of groups defined by Said Sidki

In a paper in 1982, Said Sidki defined a 2-parameter family of finitely presented groups Y(m,n) that generalise the Carmichael presentation for a finite alternating group satisfied by its generating 3-cycles (1,2,t) for t⩾3. For m⩾2 and n⩾2, the group Y(m,n) is the abstract group generated by elements a1,a2,⋯,am subject to the defining relations ain=1 for 1⩽i⩽m and (aikajk)2=1 for 1⩽i<j⩽m and 1⩽k⩽[n2]. Sidki investigated the structure of various subfamilies of these groups, for small values of m or n, and has conjectured that they are all finite. Sidki's conjecture remains open. In this paper it is shown that for all m⩾3, the group Y(m,6) is finite, and is isomorphic to a semi-direct product of an elementary abelian 2-group of order 2m(m+3)/2 by Y(m,3)≅Am+2. Also we exploit a computation for the group Y(3,8) to prove that Y(m,8) is a finite 2-group, for all m.

THE CENTRAL SUBGROUPS OF THE NONABELIAN TENSOR SQUARES OF SOME BIEBERBACH GROUPS WITH ELEMENTARY ABELIAN 2-GROUP POINT GROUP

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.

On the probability of being a deficient square group on 2-element subsets

ABSTRACTTwo elements x and y of a group G satisfy the deficient square property on 2-subsets if |{x,y}2|<4. Let ds(G) be the probability that two randomly chosen elements x and y of G satisfy the deficient square property, that is, xy = yx or . Freiman in 1981 showed that ds(G) = 1 for a finite group G if and only if G is a direct product of the quaternion group of order 8 with an elementary abelian 2-group. We show that if ds(G)<1, then , with equality if and only if G is a direct product of the dihedral group of order 8 with an elementary abelian 2-group.

On groups with automorphisms whose fixed points are Engel

We study finite and profinite groups admitting an action by an elementary abelian group under which the centralizers of automorphisms consist of Engel elements. In particular, we prove the following theorems. Let q be a prime and A an elementary abelian q-group of order $$q^2$$ acting coprimely on a profinite group G. Assume that all elements in $$C_{G}(a)$$ are Engel in G for each $$a\in A^{\#}$$ . Then, G is locally nilpotent. Let q be a prime, n a positive integer and A an elementary abelian group of order $$q^3$$ acting coprimely on a finite group G. Assume that for each $$a\in A^{\#}$$ every element of $$C_{G}(a)$$ is n-Engel in $$C_{G}(a)$$ . Then, the group G is k-Engel for some $$\{n,q\}$$ -bounded number k.

A decomposition of the Brauer–Picard group of the representation category of a finite group

We present an approach of calculating the group of braided autoequivalences of the category of representations of the Drinfeld double of a finite dimensional Hopf algebra H and thus the Brauer–Picard group of H-mod. We consider two natural subgroups and a subset as candidates for generators. In this article H is the group algebra of a finite group G. As our main result we prove that any element of the Brauer–Picard group, fulfilling an additional cohomological condition, decomposes into an ordered product of our candidates.For elementary abelian groups G our decomposition reduces to the Bruhat decomposition of the Brauer–Picard group, which is in this case a Lie group over a finite field. Our results are motivated by and have applications to symmetries and defects in 3d-TQFT and group extensions of fusion categories.

On the Loewy length of the center of a block with elementary abelian defect groups

ABSTRACTLet G be a finite group and k an algebraically closed field of characteristic p. In this paper we investigate the Loewy structure of centers of indecomposable group algebras kG, for groups G with a normal elementary abelian Sylow p-subgroup. Furthermore, we show a reduction result for the case that a normal abelian Sylow p-subgroup is acted upon by a subgroup of its automorphism group; this is fundamental in providing generic formulae for the Loewy lengths considered.

On the Brauer–Picard groups of fusion categories

We develop methods of computation of the Brauer–Picard groups of fusion categories and apply them to compute such groups for several classes of fusion categories of prime power dimension: representation categories of elementary abelian groups with twisted associativity, extra special p-groups, and the Kac–Paljutkin Hopf algebra. We conclude that many finite groups of Lie type occur as composition factors of the Brauer–Picard groups of pointed fusion categories.

Algebraic games—playing with groups and rings

Two players alternate moves in the following impartial combinatorial game: Given a finitely generated abelian group A, a move consists of picking some $$0 \ne a \in A$$ . The game then continues with the quotient group $$A/\langle a \rangle $$ . We prove that under the normal play rule, the second player has a winning strategy if and only if A is a square, i.e. $$A \cong B \times B$$ for some abelian group B. Under the misere play rule, only minor modifications concerning elementary abelian groups are necessary to describe the winning situations. We also compute the nimbers, i.e. Sprague–Grundy values of 2-generated abelian groups. An analogous game can be played with arbitrary algebraic structures. We study some examples of non-abelian groups and commutative rings such as R[X], where R is a principal ideal domain.

2-Arc-Transitive Regular Covers of Kn,n Having the Covering Transformation Group ℤp2

This paper contributes to the classification of finite 2-arc-transitive graphs. In [12], all the regular covers of complete bipartite graphs Kn,n were classified, whose covering transformation group is cyclic and whose fibre-preserving automorphism group acts 2-arc-transitively. In this paper, a further classification is achieved for all the regular covers of Kn,n, whose covering transformation group is elementary abelian group of order p2 and whose fibre-preserving automorphism group acts 2-arc-transitively. As a result, two new infinite families of 2-arc-transitive graphs are found. Moveover, it will be explained that it seems to be infeasible to classify all such covers when the covering transformation group is an elementary abelian group of order pk for an arbitrary integer k.

Almost split sequences for polynomial GrT-modules and polynomial parts of Auslander–Reiten components

In [8], Doty, Nakano and Peters defined infinitesimal Schur algebras, combining the approach via polynomial representations with the approach via GrT-modules to representations of the algebraic group G=GLn. We study analogues of these algebras and their Auslander–Reiten theory for reductive algebraic groups G and Borel subgroups B by considering the categories of polynomial representations of GrT and BrT as full subcategories of modGrT and modBrT, respectively. We show that every component Θ of the stable Auslander–Reiten quiver Γs(GrT) of modGrT whose constituents have complexity 1 contains only finitely many polynomial modules. For G=GL2,r=1 and T⊆G the torus of diagonal matrices, we identify the polynomial part of the stable Auslander–Reiten quiver of GrT and use this to determine the Auslander–Reiten quiver of the infinitesimal Schur algebras in this situation. For the Borel subgroup B of lower triangular matrices of GL2, the category of BrT-modules is related to representations of elementary abelian groups of rank r. In this case, we can extend our results about modules of complexity 1 to modules of higher Frobenius kernels arising as outer tensor products.

On eight-dimensional unital division algebras over finite fields

ABSTRACTThe purpose of this paper is to investigate the eight-dimensional non-associative unital division algebras {0}≠A over finite fields, admitting elementary abelian 2-group of automorphisms E≤Aut(A), of order 8, using tools from algebraic geometry.

Erratum to: A characterization of elementary abelian 2-groups

Erratum to: A characterization of elementary abelian 2-groups

Lannes’ T functor on injective unstable modules and Harish-Chandra restriction

In the 1980's, the magic properties of the cohomology of elementary abelian groups as modules over the Steenrod algebra initiated a long lasting interaction between topology and modular representation theory in natural characteristic. The Adams-Gunawardena-Miller theorem in particular, showed that their decomposition is governed by the modular representations of the semi-groups of square matrices. Applying Lannes' T functor on the summands L P := Hom Mn(Fp) (P, H * (F p) n) defines an intriguing construction in representation theory. We show that T(L P) ∼ = L P ⊕ H * V 1 ⊗ L δ(P) , defining a functor δ from F p [M n (F p)]-projectives to F p [M n−1 (F p)]-projectives. We relate this new functor δ to classical constructions in the representation theory of the general linear groups.

Pauli gradings on Lie superalgebras and graded codimension growth

We introduce grading on certain finite dimensional simple Lie superalgebras of type P(t) by elementary abelian 2-group. This grading gives rise to Pauli matrices and is a far generalization of (Z2×Z2)-grading on Lie algebra of (2×2)-traceless matrices.We use this grading for studying numerical invariants of polynomial identities of Lie superalgebras. In particular, we compute graded PI-exponent corresponding to Pauli grading.

Representations of finite group schemes and morphisms of projective varieties

Given a finite group scheme $\cG$ over an algebraically closed field $k$ of characteristic $\Char(k)=p>0$, we introduce new invariants for a $\cG$-module $M$ by associating certain morphisms $\deg^j_M : U_M \lra \Gr_d(M) \ \ (1\!\le\!j\!\le\! p\!-\!1)$ to $M$ that take values in Grassmannians of $M$. These maps are studied for two classes of finite algebraic groups, infinitesimal group schemes and elementary abelian group schemes. The maps associated to the so-called modules of constant $j$-rank have a well-defined degree ranging between $0$ and $j\rk^j(M)$, where $\rk^j(M)$ is the generic $j$-rank of $M$. The extreme values are attained when the module $M$ has the equal images property or the equal kernels property. We establish a formula linking the $j$-degrees of $M$ and its dual $M^\ast$. For a self-dual module $M$ of constant Jordan type this provides information concerning the indecomposable constituents of the pull-back $\alpha^\ast(M)$ of $M$ along a $p$-point $\alpha : k[X]/(X^p) \lra k\cG$.