Order Q2 Research Articles

On finite groups with automorphisms whose fixed points are Engel

The main result of the paper is the following theorem. Let q be a prime, n a positive integer, and A an elementary abelian group of order q2. Suppose that A acts coprimely on a finite group G and assume that for each \({a \in A^{\#}}\) every element of CG(a) is n-Engel in G. Then the group G is k-Engel for some \({\{n,q\}}\)-bounded number k.

Existence of canonically inherited arcs in Moulton planes of odd order

We address the problem of determining the spectrum of all possible values of k for canonically inherited k-arcs in Moulton planes of odd order q2, with q≡3(mod4), arising from hyperbole in AG(2,q2).

Hermitian Self-Dual Abelian Codes

Hermitian self-dual abelian codes in a group ring F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> [G], where F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> is a finite field of order q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> and G is a finite abelian group, are studied. Using the well-known discrete Fourier transform decomposition for a semisimple group ring, a characterization of Hermitian self-dual abelian codes in F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> [G] is given, together with an alternative proof of necessary and sufficient conditions for the existence of such a code in F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> [G], i.e., there exists a Hermitian self-dual abelian code in F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> [G] if and only if the order of G is even and q = 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</sup> for some positive integer l. Later on, the study is further restricted to the case where F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2l</sup> [G] is a principal ideal group ring, or equivalently, G ≅ A⊕Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> k with 2 ≠ |A|. Based on the characterization obtained, the number of Hermitian self-dual abelian codes in F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> 2l [A⊕Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> k] can be determined easily. When A is cyclic, this result answers an open problem of Jia et al. concerning Hermitian self-dual cyclic codes. In many cases, F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> 2l [A⊕Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> k] contains a unique Hermitian self-dual abelian code. The criteria for such cases are determined in terms of l and the order of A. Finally, the distribution of finite abelian groups A such that a unique Hermitian self-dual abelian code exists in F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> 2l [A ⊕ Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> ] is established, together with the distribution of odd integers m such that a unique Hermitian self-dual cyclic code of length 2 m over F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> 2l exists.

The physical squeezed limit: consistency relations at order q2

In single-field models of inflation the effect of a long mode with momentum q reduces to a diffeomorphism at zeroth and first order in q. This gives the well-known consistency relations for the n-point functions. At order q2 the long mode has a physical effect on the short ones, since it induces curvature, and we expect that this effect is the same as being in a curved FRW universe. In this paper we verify this intuition in various examples of the three-point function, whose behaviour at order q2 can be written in terms of the power spectrum in a curved universe. This gives a simple alternative understanding of the level of non-Gaussianity in single-field models. Non-Gaussianity is always parametrically enhanced when modes freeze at a physical scale kph, f shorter than H: fNL ∼ (kph, f/H)2.

On PGL(2,q)-invariant unitals embedded in Desarguesian or in Hughes planes

In this paper it is shown that any unital embedded either in PG(2,q2) or in the Hughes plane of order q2, which is invariant under PGL(2,q)×〈σ〉, where σ is the Frobenius involution, is Hermitian or classical Rosati, respectively.

A combinatorial characterization of the Hermitian surface

In this paper we prove that in a projective space of dimension three and square order q2 a (q5+q3+q2+1)-set of class [1,2,…,q+1,q2+1]1 and type (m,n)2 is a Hermitian surface.

Relative hemisystems on the Hermitian surface

Let S be a generalized quadrangle of order (q 2,q) containing a subquadrangle S′ of order (q,q). Then any line of S either meets S′ in q+1 points or is disjoint from S′. After Penttila and Williford (J. Comb. Theory, Ser. A 118:502–509, 2011), we call a subset H of the lines disjoint from S′ a relative hemisystem of S with respect to S′, provided that for each point x of S∖S′, exactly half of the lines through x disjoint from S′ lie in H. A new infinite family of relative hemisystems on the generalized quadrangle $\mathcal{H}(3,q^{2})$ admitting the linear group PSL(2,q) as an automorphism group is constructed. The association schemes arising from our construction are not equivalent to those arising from the Penttila–Williford relative hemisystems.

A characterization of the non-Baer lines of a Hermitian surface

We give a characterization of the set of the lines which either belong to or are tangent to a non-singular Hermitian surface in the projective space of dimension 3 and order q 2.

A group theoretic characterization of Buekenhout–Metz unitals in [formula omitted] containing conics

Let U be a unital in PG(2,q2), q=ph and let G be the group of projectivities of PG(2,q2) stabilizing U. In this paper we prove that U is a Buekenhout–Metz unital containing conics and q is odd if, and only if, there exists a point A of U such that the stabilizer of A in G contains an elementary Abelian p-group of order q2 with no non-identity elations.

Factorized spread sets and translation planes with large homology groups

Abstract In this note we construct some series of translation planes of order q 2 which possess two commuting homology groups of order q – 1 and (q + 1)(q – 1, 2) respectively. These translation planes can be considered as relatives of nearfield planes: for nearfield planes the nontrivial part of the associated spread set is a group whereas in our case this set is the product of two groups.

HMO-planes

Abstract Hiramine, Matsumoto and Oyama [Osaka J. Math. 24: 123–137, 1987] made the remarkable discovery that every translation plane of order q 2 that is 2-dimensional over its kernel produces translation planes of order q 4. This construction is studied, with emphasis on isomorphisms among the resulting planes.

Classification of Projective Translation Planes of Order q2 Admitting a Two-Transitive Orbit of Length q + 1

A classification is given of all projective translation planes of order q2 that admit a collineation group G admitting a two-transitive orbit of q + 1 points.

Lifting Subregular Spreads

Let S k denote a set of k reguli in a Desarguesian affine plane $$\sum_{q^2}$$ of order q 2. It is shown that, for every odd integer s > 1, there is a corresponding set S s k of k reguli in any Desarguesian plane $$\sum_{q^{2s}}$$ of order q 2s such the line intersection properties of the reguli of S s k are inherited from those of S k . Hence, sets of mutually disjoint reguli in $$\sum_{q^2}$$ ‘lift’ to sets of mutually disjoint reguli in $$\sum_{q^{2s}}$$ . Thus, the existence of a subregular spread in PG(3, q) produces an infinite class of subregular spreads in spaces PG(3, q s ).

Semifield planes of order q4 with kernel Fq2 and center Fq

A classification of semifield planes of order q4 with kernel Fq2 and center Fq is given. For q an odd prime, this proves the conjecture stated in [M. Cordero, R. Figueroa, On the semifield planes of order 54 and dimension 2 over the kernel, Note Mat. (in press)]. Also, we extend the classification of semifield planes lifted from Desarguesian planes of order q2, q odd, obtained in [M. Cordero, R. Figueroa, On some new classes of semifield planes, Osaka J. Math. 30 (1993) 171–178], to the even characteristic case.

A New Method for Constructing Williamson Matrices

For every prime power q ? 1 (mod 4) we prove the existence of (q; x, y)-partitions of GF(q) with q=x2+4y2 for some x, y, which are very useful for constructing SDS, DS and Hadamard matrices. We discuss the transformations of (q; x,y)-partitions and, by using the partitions, construct generalized cyclotomic classes which have properties similar to those of classical cyclotomic classes. Thus we provide a new construction for Williamson matrices of order q2.

Nuclear fusion in finite semifield planes

Abstract We study the subplanes of finite semifield planes that are coordinatizable by subfields F of some semifield D such that F lies in at least two of the three seminuclear fields N ℓ(D), Nm (D), and Nr (D). Our main results determine completely the combinatorial configurations associated with such subplanes, and enables, for example, a computational method to determine the number of nuclear planes of order q in semifield planes of order qt . The results have a number of applications. Firstly, they imply ‘fusion’ theorems. The most basic one is that if two or more seminuclear subfields, of a semifield D coordinatizing a translation plane π, are each isomorphic to GF(q), then π may be recoordinatized by a semifield E such that the indicated seminuclear fields, of order q, coincide. The most important case is nuclear fusion: if all three seminuclei of D are isomorphic to GF(q) then the nucleus N(E) ≅ GF(q). Further applications are concerned with semifield spreads π of order q 2. We classify all such π that admit three homology groups of order q – 1 with dierent axis (the shears axis, the infinite line, and any other component), thus generalizing a theorem of D. E. Knuth, who proved an algebraic version of the result.

Transversal-Free Translation Nets

Translation planes of order q2 containing non-Desarguesian Baer subplanes are used to construct transversal-free translation nets with very small deficiencies. Also, a generalization of the ideas of Bruen shows that any non-Desarguesian spread in PG(3, q) produces a transversal-free net of small deficiency.

Authenticated Key Exchange Provably Secure Against the Man-in-the-Middle Attack

The standard Diffie--Hellman key exchange is suseptible to an attack known as the man-in-the-middle attack. Lack of authentication in the protocol makes this attack possible. Adding separate authentication to the protocol solves the problem but adds extra transmission and computation costs. Protocols which combine the authentication with the key exchange (an authenticated key exchange) are more efficient but until now none were provably secure against the man-in-the-middle attack. This paper describes an authenticated key exchange based on the difficulty of the q th-root problem, a problem believed to be equivalent to the discrete logarithm problem over groups of order q2 (where q is a large prime) and parallel to the square-root problem over the ring modulo N , where N is a strong two prime composite integer. We show that mounting a man-in-the-middle attack for our protocol is equivalent to breaking the Diffie--Hellman problem in the group.

A Result on Spreads of the Generalized QuadrangleT2(O), withOan Oval Arising from a Flock, and Applications

If F is a flock of the quadratic cone K of PG (3, q), q even, then the corresponding generalized quadrangle S(F) of order (q2,q ) has subquadrangles T2(O), with O an oval, of order q. We prove in a geometrical way that any such T2(O) has spreads S consisting of an element y∈O and the q2lines not in the plane PG (2,q ) of O of q quadratic cones Kx,x∈O− { y }, of the space PG (3, q) containing T2(O), where Kxhas vertex x, is tangent to PG (2, q) at xy and has nucleus line xn, with n the nucleus of O. We also show how the oval O can be directly constructed from the flock F.

Subquadrangles of Generalized Quadrangles of Order (q2, q), q Even

First, we survey the known generalized quadrangles of order (q2, q), q even, including a description of their known subquadrangles of order q. Then, in the case of Tits' generalized quadrangles, we completely classify the subquadrangles of order q, while in the case of the flock quadrangles we classify the subquadrangles of order q which contain the base point. Finally, we determine the full automorphism groups of the Tits generalized quadrangles T2(O) of order q and T3(Ω) of order (q, q2).