Elementary Abelian Group Of Order Research Articles

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.

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

On a Huge Family of Non-Schurian Schur Rings

In his famous monograph on permutation groups, H. Wielandt gives an example of a Schur ring over an elementary abelian group of order $p^2$ ($p>3$ is a prime), which is non-schurian, that is, it is the transitivity module of no permutation group. Generalizing this example, we construct a huge family of non-schurian Schur rings over elementary abelian groups of even rank.

Diagonal groups and arcs over groups

In an earlier paper by three of the present authors and Csaba Schneider, it was shown that, for mge 2, a set of m+1 partitions of a set Omega , any m of which are the minimal non-trivial elements of a Cartesian lattice, either form a Latin square (if m=2), or generate a join-semilattice of dimension m associated with a diagonal group over a base group G. In this paper we investigate what happens if we have m+r partitions with rge 2, any m of which are minimal elements of a Cartesian lattice. If m=2, this is just a set of mutually orthogonal Latin squares. We consider the case where all these squares are isotopic to Cayley tables of groups, and give an example to show the groups need not be all isomorphic. For m>2, things are more restricted. Any m+1 of the partitions generate a join-semilattice admitting a diagonal group over a group G. It may be that the groups are all isomorphic, though we cannot prove this. Under an extra hypothesis, we show that G must be abelian and must have three fixed-point-free automorphisms whose product is the identity. (We describe explicitly all abelian groups having such automorphisms.) Under this hypothesis, the structure gives an orthogonal array, and conversely in some cases. If the group is cyclic of prime order p, then the structure corresponds exactly to an arc of cardinality m+r in the (m-1)-dimensional projective space over the field with p elements, so all known results about arcs are applicable. More generally, arcs over a finite field of order q give examples where G is the elementary abelian group of order q. These examples can be lifted to non-elementary abelian groups using p-adic techniques.

The first example of a simple [formula omitted] design

We give the very first example of a simple 2−(81,6,2) design. Its points are the elements of the elementary abelian group of order 81 and each block is the union of two parallel lines of the 4-dimensional geometry over the field of order 3. Hence it is also additive.

The Noether number of p-groups

A group of order [Formula: see text] ([Formula: see text] prime) has an indecomposable polynomial invariant of degree at least [Formula: see text] if and only if the group has a cyclic subgroup of index at most [Formula: see text] or it is isomorphic to the elementary abelian group of order 8 or the Heisenberg group of order 27.

FITTING SUBGROUP AND NILPOTENT RESIDUAL OF FIXED POINTS

Let $q$ be a prime and let $A$ be an elementary abelian group of order at least $q^{3}$ acting by automorphisms on a finite $q^{\prime }$-group $G$. We prove that if $|\unicode[STIX]{x1D6FE}_{\infty }(C_{G}(a))|\leq m$ for any $a\in A^{\#}$, then the order of $\unicode[STIX]{x1D6FE}_{\infty }(G)$ is $m$-bounded. If $F(C_{G}(a))$ has index at most $m$ in $C_{G}(a)$ for any $a\in A^{\#}$, then the index of $F_{2}(G)$ is $m$-bounded.

Cohomology of symplectic groups and Meyer’s signature theorem

Meyer showed that the signature of a closed oriented surface bundle over a surface is a multiple of $4$, and can be computed using an element of $H^2(\mathsf{Sp}(2g, \mathbb{Z}),\mathbb{Z})$. Denoting by $1 \to \mathbb{Z} \to \widetilde{\mathsf{Sp}(2g,\mathbb{Z})} \to \mathsf{Sp}(2g,\mathbb{Z}) \to 1$ the pullback of the universal cover of $\mathsf{ Sp}(2g,\mathbb{R})$, Deligne proved that every finite index subgroup of $\widetilde{\mathsf {Sp}(2g, \mathbb{Z})}$ contains $2\mathbb{Z}$. As a consequence, a class in the second cohomology of any finite quotient of $\mathsf{Sp}(2g, \mathbb{Z})$ can at most enable us to compute the signature of a surface bundle modulo $8$. We show that this is in fact possible and investigate the smallest quotient of $\mathsf{Sp}(2g, \mathbb{Z})$ that contains this information. This quotient $\mathfrak{H}$ is a non-split extension of $\mathsf {Sp}(2g,2)$ by an elementary abelian group of order $2^{2g+1}$. There is a central extension $1\to \mathbb{Z}/2\to\tilde{{\mathfrak{H}}}\to\mathfrak{H}\to 1$, and $\tilde{\mathfrak{H}}$ appears as a quotient of the metaplectic double cover $\mathsf{Mp}(2g,\mathbb{Z})=\widetilde{\mathsf{Sp}(2g,\mathbb{Z})}/2\mathbb{Z}$. It is an extension of $\mathsf{Sp}(2g,2)$ by an almost extraspecial group of order $2^{2g+2}$, and has a faithful irreducible complex representation of dimension $2^g$. Provided $g\ge 4$, $\widetilde{\mathfrak{H}}$ is the universal central extension of $\mathfrak{H}$. Putting all this together, we provide a recipe for computing the signature modulo $8$, and indicate some consequences.

Bounds on the number of ideals in finite commutative nilpotent $\mathbb{F}_p$-algebras

Let $A$ be a finite commutative nilpotent $\mathbb{F}_p$-algebra structure on $G$, an elementary abelian group of order $p^n$. If $K/k$ is a Galois extension of fields with Galois group $G$ and $A^p = 0$, then corresponding to $A$ is an $H$-Hopf Galois structure on $K/k$ of type $G$. For that Hopf Galois structure we may study the image of the Galois correspondence from $k$-subHopf algebras of $H$ to subfields of $K$ containing $k$ by utilizing the fact that the intermediate subfields correspond to the $\mathbb{F}_p$-subspaces of $A$, while the subHopf algebras of $H$ correspond to the ideals of $A$. We obtain upper and lower bounds on the proportion of subspaces of $A$ that are ideals of $A$, and test the bounds on some examples.

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 $\mathbb{F}_{q^2}$.

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.

Locally 3-Arc-Transitive Regular Covers of Complete Bipartite Graphs

In this paper, locally $3$-arc-transitive regular covers of complete bipartite graphs are studied, and results are obtained that apply to arbitrary covering transformation groups. In particular, methods are obtained for classifying the locally $3$-arc-transitive graphs with a prescribed covering transformation group, and these results are applied to classify the locally $3$-arc-transitive regular covers of complete bipartite graphs with covering transformation group isomorphic to a cyclic group or an elementary abelian group of order $p^2$.

GROUPS OF ORDER 16 AS GALOIS GROUPS OVER THE 2-ADIC NUMBERS

Let K be a Galois extension of the 2-adic numbers Q2 of degree 16 and let G be the Galois group of K/Q2. We show that G can be determined by the Galois groups of the octic subfields of K. We also show that all 14 groups of order 16 occur as the Galois group of some Galois extension K/Q2 except for E16, the elementary abelian group of order 2 4 . For the other 13 groups G, we give a degree 16 polynomial f(x) such that the Galois group of f over Q2 is G.

The automorphism group of a self-dual $[72,36,16]$ code does not contain $\mathcal S_3$, $\mathcal A_4$ or $D_8$

A computer calculation with $M$AGMA shows that there is noextremal self-dual binary code $\mathcal{C}$ of length $72$ whoseautomorphism group contains the symmetric group of degree $3$, thealternating group of degree $4$ or the dihedral group of order $8$.Combining this with the known results in the literature one obtainsthat $Aut(\mathcal{C})$ has order at most $5$ or isisomorphic to the elementary abelian group of order $8$.

The automorphism group of a self-dual [formula omitted] code is not an elementary abelian group of order 8

The existence of an extremal self-dual binary linear code C of length 72 is a long-standing open problem. We continue the investigation of its automorphism group: looking at the combination of the subcodes fixed by different involutions and doing a computer calculation with Magma, we prove that Aut(C) is not isomorphic to the elementary abelian group of order 8. Combining this with the known results in the literature one obtains that Aut(C) has order at most 5.

The Automorphism Group of an Extremal $[{72,36,16}]$ Code Does Not Contain $Z_{7}$, $Z_{3}\times Z_{3}$, or $D_{10}$

A computer calculation with Magma shows that there is no extremal self-dual binary code C of length 72 that has an automorphism group containing either the dihedral group of order 10, the elementary abelian group of order 9, or the cyclic group of order 7. Combining this with the known results in the literature, one obtains that the order of Aut(C) is either 5 or divides 24.

Fixed points of coprime operator groups

Let m be a positive integer and A an elementary abelian group of order q r with r ⩾ 2 acting on a finite q ′ -group G. We show that if for some integer d such that 2 d ⩽ r − 1 the dth derived group of C G ( a ) has exponent dividing m for any a ∈ A # , then G ( d ) has { m , q , r } -bounded exponent and if γ r − 1 ( C G ( a ) ) has exponent dividing m for any a ∈ A # , then γ r − 1 ( G ) has { m , q , r } -bounded exponent.

A check digit system for hexadecimal numbers

We present a new check digit system for hexadecimal numbers which is based on the use of a suitable automorphism of the elementary abelian group of order sixteen. Our system is able to detect all s...

On the rank two geometries of the groups PSL(2, q): part I

We determine all firm and residually connected rank 2 geometries on which PSL(2, q ) acts flag-transitively, residually weakly primitively and locally two-transitively, where one of the maximal parabolic subgroups is isomorphic to E q : ( q − 1)/(2, q − 1), where E q denotes an elementary abelian group of order q, or D 2 n ( q ) , the dihedral group of order 2 n ( q ) where n ( q ) := ( q ± 1)/gcd(2, q − 1) for some prime-power q .