The Semifields of Order q 4 Where q is a Power of 2

The aim of this article is to investigate all semifields A of order q ⁴ where q is a power of 2, having the properties that A contains GF(q ²) in its left nucleus and admits a four-group E ? Z ₂ × Z ₂ of automorphisms of A acting freely on A.     

On the Automorphism Group of a Semifield of Order 54

The purpose of this article is to determine Aut(A) where A is a semifield of order 5⁴ admitting an automorphism group E ? Z ₂ × Z ₂ acting freely on A.     

The Autotopism Group of A Semifield of Order p 4, p is an Odd Prime

The aim of this article is to investigate the autotopism group of a semifield of order p ⁴, p is an odd prime, admitting a four-group of automorphisms E? Z ₂ × Z ₂ acting freely on A.     

The Multiplicity Free Permutation Representations of the Ree Groups 2 G 2(q), the Suzuki Groups 2 B 2(q), and Their Automorphism Groups

Abstract

Let G a simple group of type ² B ₂(q) or ² G ₂(q), where q is an odd power of 2 or 3, respectively. The main goal of this paper is to determine the multiplicity free permutation representations of G and A ≤ Aut(G) where A is a subgroup containing a copy of G. Let B be a Borel subgroup of G. If G = ² B ₂(q) we show that there is only one non-trivial multiplicity free permutation representation, namely the representation of G associated to the action on G/B. If G = ² G ₂(q) we show that there are exactly two such non-trivial representations, namely the representations of G associated to the action on G/B and the action on G/M, where M = UC with U the maximal unipotent subgroup of B and C the unique subgroup of index 2 in the maximal split torus of B. The multiplicity free permutation representations of A correspond to the actions on A/H where H is isomorphic to a subgroup containing B if G = ² B ₂(q), and containing M if G = ² G ₂(q). The problem of determining the multiplicity free representations of the finite simple groups is important, for example, in the classification of distance-transitive graphs.     

Cubic surfaces in AG(3, q) and projective planes of order q 3

Spread sets of projective planes of order q ³ are represented as sets of q ³ points in A AG(3, q ³). A line through the origin in A can be interpreted as a space A ₀ AG(3, q), and the spread set induces a cubic surface L in A ₀. If the projective plane is a semifield plane of dimension 3 over its kernel, then L has the property that it misses a plane of A ₀. Determining all such surfaces L leads to a complete classification of the semifield planes of order q ³, whose spread sets are division algebras of dimension 3.An alternative proof of a result due to Menichetti, that finite division algebras of dimension 3 are associative or are twisted fields, follows with the classification.     

On the existence of semifields of prime power order admitting free automorphism groups

The purpose of this paper is to prove the existence of semifields of order q ⁴ for any odd prime power q = p^r, q > 3, admitting a free automorphism group isomorphic to Z ₂ × Z ₂.     

On the Validity of Thompson's Conjecture for Finite Simple Groups

In this article, we prove a conjecture of J. G. Thompson for the finite simple group ² D _n (q). More precisely, we show that every finite group G with the property Z(G) = 1 and N(G) = N(² D _n (q)) is necessarily isomorphic to ² D _n (q). Note that N(G) is the set of lengths of conjugacy classes of G.     

Generic Modules Over a Class of Drinfeld's Quantum Doubles

Let k be a field and A _n (ω) be the Taft's n ²-dimensional Hopf algebras. When n is odd, the Drinfeld quantum double D(A _n (ω)) of A _n (ω) is a Ribbon Hopf algebra. In the previous articles, we constructed an n ⁴-dimensional Hopf algebra H _n (p, q) which is isomorphic to D(A _n (ω)) if p ≠ 0 and q = ω^?1, and studied the finite dimensional representations of H _n (1, q). We showed that the basic algebra of any nonsimple block of H _n (1, q) is independent of n. In this article, we examine the infinite representations of H ₂(1, ? 1), or equivalently of H _n (1, q)?D(A _n (ω)) for any n ≥ 2. We investigate the indecomposable and algebraically compact modules over H ₂(1, ? 1), describe the structures of these modules and classify them under the elementary equivalence.     

Characterisations of A p+2 and L 2(2 q )

Let G be a transitive group of degree p+2 with p|G| where p ≧ 5 is a prime number, then (i) G is isomorphic to S _p+2 or A _p+2, if G has an element of order 4, (ii) G is isomorphic to L ₂(2 ^q ) or P Γ L ₂ (2 ^q ), if 2 ^q − 1=p is a Mersenne prime and G has no element of order 4.     

Hochschild Cohomology of Quiver Algebras Defined by Two Cycles and a Quantum-Like Relation

We consider quiver algebras A _q over a field k defined by two cycles and a quantum-like relation depending on a nonzero element q in k, and describe the minimal projective bimodule resolution of A _q . In particular, in the case q = 1, we determine the Hochschild cohomology ring of A ₁ and show that it is a finitely generated k-algebra. Moreover the Hochschild cohomology ring of A ₁ modulo nilpotence is isomorphic to the polynomial ring of two variables.     

Writing Elements of PSL(2, q) as Commutators

We prove that for q ≥ 13, an element A of SL(2, q) is the commutator of a generating pair if and only if A ≠ ?I and the trace of A is not 2. Consequently, when q is odd and q ≥ 13, every nontrivial element of PSL(2, q) is the commutator of a generating pair, and when q is even, an element of PSL(2, q) is the commutator of a generating pair if and only if its trace is not 0. The proof of these results also leads to an improved lower bound on the number of T-systems of generating pairs of PSL(2, q).     

Classification of triply-retractive semifields of order q 4

A complete classification of the semifield spreads of order q⁴ that triply-retract to mixed partitions of PG (3, q²) is given.     

On translation planes of orderq 2 that admit a group of orderq 2 (q−1); bartolone's theorem

This article is concerned with translation planesP of orderq ² and kernelK isomorphic toG F(q). IfP admits a collineation groupG in the linear translation. complement and the order ofG K/K isq ²(q?1) then it is shown thatP is either a semifield plane or is a Lüneburg-Tits, Walker or Betten plane. This generalizes earlier work of Bartolone.     

Some Maximal Function Fields and Additive Polynomials

We derive explicit equations for the maximal function fields F over 𝔽 _{q ²ⁿ} given by F = 𝔽 _{q ²ⁿ} (X, Y) with the relation A(Y) = f(X), where A(Y) and f(X) are polynomials with coefficients in the finite field 𝔽 _{q ²ⁿ} , and where A(Y) is q-additive and deg(f) = q ⁿ  + 1. We prove in particular that such maximal function fields F are Galois subfields of the Hermitian function field H over 𝔽 _{q ²ⁿ} (i.e., the extension H/F is Galois).     

Classi di isomorfismo delle cubiche diF q

LetF _q (q=p^r) be a field of characteristicp>3 andA the set of all elliptic cubic curves overF _q having a given absolute invariantj. Furthermore let ≈be the following equivalence relation: « if and only if and F_q are isomorphic overF _q as abelian varieties». The aim of this paper is to study the equivalence classes inA, induced by ≈, and the Frobenius' traces of the cubic curves belonging to different subclasses ofA.     

Transitive A 6-invariant k-arcs in PG(2, q)

For q = p ^r with a prime p ≥ 7 such that ${q \equiv 1}$ or 19 (mod 30), the desarguesian projective plane PG(2, q) of order q has a unique conjugacy class of projectivity groups isomorphic to the alternating group A ₆ of degree 6. For a projectivity group ${\Gamma \cong A_6}$ of PG(2, q), we investigate the geometric properties of the (unique) Γ-orbit ${\mathcal{O}}$ of size 90 such that the 1-point stabilizer of Γ in its action on ${\mathcal O}$ is a cyclic group of order 4. Here ${\mathcal O}$ lies either in PG(2, q) or in PG(2, q ²) according as 3 is a square or a non-square element in GF(q). We show that if q ≥ 349 and q ≠ 421, then ${\mathcal O}$ is a 90-arc, which turns out to be complete for q = 349, 409, 529, 601,661. Interestingly, ${\mathcal O}$ is the smallest known complete arc in PG(2,601) and in PG(2,661). Computations are carried out by MAGMA.     

A characterization of PGL(2,q),q ODD

LetG be a finite group andA andB solvable subgroups ofG, such thatG=AB and 2 is the only common prime divisor ofA andB. Under suitable restrictions of the 2-structure ofA andB, it is shown that eitherG is solvable orG contains a solvable normal subgroupN, such thatG/N contains a normal subgroup, which is isomorphic to PGL(2,q),q odd.