Related papers: Singer 8-arcs of Mathon type in PG(2,2^7)

An arc in $\Z^2_n$ is defined to be a set of points no three of which are collinear. We describe some properties of arcs and determine the maximum size of arcs for some small $n$.

Given an octonion algebra over a field k, its automorphism group G is an algebraic semisimple k-group of type G_2. We study the maximal tori of G in terms of the algebra C.

We give necessary and sufficient conditions for an orthogonal group defined over a field of characteristic not 2 to contain a maximal torus of a given type.

For a prime number p, we characterize the groups that may arise as torsion subgroups of an elliptic curve with complex multiplication defined over a number field of degree 2p. In particular, our work shows that a classification in the…

Let $G$ be a finite almost simple group. It is well known that $G$ can be generated by 3 elements, and in previous work we showed that 6 generators suffice for all maximal subgroups of $G$. In this paper we consider subgroups at the next…

A numerical semigroup is called cyclotomic if its corresponding numerical semigroup polynomial $P_S(x)=(1-x)\sum_{s\in S}x^s$ is expressable as the product of cyclotomic polynomials. Ciolan, Garc\'ia-S\'anchez, and Moree conjectured that…

Explicit constructions of infinite families of scattered ${\mathbb F}_q$--linear sets in $PG(r-1,q^t)$ of maximal rank $\frac{rt}2$, for $t$ even, are provided. When $q=2$ and $r$ is odd, these linear sets correspond to complete caps in…

An upper bound on degrees of elements of a minimal generating system for invariants of quivers of dimension (2,...,2) is established over a field of arbitrary characteristic and its precision is estimated. The proof is based on the…

In this article, we prove that if $\mathcal{D}$ is a $2$-design with $k=7$ admitting flag-transitive almost simple automorphism group with socle an alternating group, then $\mathcal{D}$ is $PG_{2}(3,2)$ with parameter set $(15,7,3)$ and…

We study the types of non-integrable $\mathrm{G}$-structures on Riemannian manifolds. In particular, geometric types admitting a connection with totally skew-symmetric torsion are characterized. 8-dimensional manifolds equipped with a…

We give a new computer-assisted proof of the classification of maximal subgroups of the simple group ${}^2E_6(2)$ and its extensions by any subgroup of the outer automorphism group $S_3$. This is not a new result, but no earlier proof…

In this paper we study the classification of right-angled Artin groups up to commensurability. We characterise the commensurability classes of RAAGs defined by trees of diameter 4. In particular, we prove a conjecture of Behrstock and…

We first determine all height four Gorenstein sequences beginning H=(1,4,7,...), and we show that their first differences satisfy $\Delta H_{\le j/2}$ is an O-sequence. We then study the family PGor(H) parametrizing all graded Artinian…

We present a new approach to the ideal membership problem for polynomial rings over the integers: given polynomials $f_0,f_1,...,f_n\in\Z[X]$, where $X=(X_1,...,X_N)$ is an $N$-tuple of indeterminates, are there $g_1,...,g_n\in\Z[X]$ such…

We present a periodic infinite chain of finite generalisations of the exceptional structures, including e8, the exceptional Jordan algebra (and pair), and the octonions. We demonstrate that the exceptional Jordan algebra is part of an…

Given a collection of points in the plane, classifying which subsets are collinear is a natural problem and is related to classical geometric constructions. We consider collections of points in a projective plane over a finite field such…

We enumerate the number of complex irreducible representations of each degree of general unitary groups of degree 4 over principal ideal rings of length 2.

Fiber bundles over infinite fields with non-trivial ultra-norms are considered. For them geometric wrap groups are defined and investigated. Besides fields also Cayley-Dickson algebras over fields of characteristic not equal to two are…

We present an algorithm which given a presentation of a group $G$ without 2-torsion, a solution to the word problem with respect to this presentation, and an acylindricity constant ${\kappa}$, outputs a collection of tracks in an…

Let $G$ be a finite group. Denoting by ${\rm{cd}}(G)$ the set of the degrees of the irreducible complex characters of $G$, we consider the {\it character degree graph} of $G$: this is the (simple, undirected) graph whose vertices are the…