arXiv subjects

Tomasz Popiel

Publications and source records attributed to Tomasz Popiel.

23 records · Page 2Linked to original sources

arXiv2015

Generalised polygons admitting a point-primitive almost simple group of Suzuki or Ree type

Let $G$ be a collineation group of a thick finite generalised hexagon or generalised octagon $Γ$. If $G$ acts primitively on the points of $Γ$, then a recent result of Bamberg et al. shows that $G$ must be an almost simple group of Lie type. We show that, furthermore, the minimal normal subgroup $S$ of $G$ cannot be a Suzuki group or a Ree group of type $^2\text{G}_2$, and that if $S$ is a Ree group of type $^2\text{F}_4$, then $Γ$ is (up to point--line duality) the classical Ree--Tits generalised octagon.

math.GR↗

arXiv2014

Point-primitive generalised hexagons and octagons

In 2008, Schneider and Van Maldeghem proved that if a group acts flag-transitively, point-primitively, and line-primitively on a generalised hexagon or generalised octagon, then it is an almost simple group of Lie type. We show that point-primitivity is sufficient for the same conclusion, regardless of the action on lines or flags. This result narrows the search for generalised hexagons or octagons with point- or line-primitive collineation groups beyond the classical examples, namely the two generalised hexagons and one generalised octagon admitting the Lie type groups $\mathsf{G}_2(q)$, $\,^3\mathsf{D}_4(q)$, and $\,^2\mathsf{F}_4(q)$, respectively.

math.CO↗

arXiv2014

Generalised quadrangles and transitive pseudo-hyperovals

A pseudo-hyperoval of a projective space $\PG(3n-1,q)$, $q$ even, is a set of $q^n+2$ subspaces of dimension $n-1$ such that any three span the whole space. We prove that a pseudo-hyperoval with an irreducible transitive stabiliser is elementary. We then deduce from this result a classification of the thick generalised quadrangles $\mathcal{Q}$ that admit a point-primitive, line-transitive automorphism group with a point-regular abelian normal subgroup. Specifically, we show that $\mathcal{Q}$ is flag-transitive and isomorphic to $T_2^*(\mathcal{H})$, where $\mathcal{H}$ is either the regular hyperoval of $\PG(2,4)$ or the Lunelli--Sce hyperoval of $\PG(2,16)$.

math.CO↗

arXiv2014

Nilpotent-independent sets and estimation in matrix algebras

Efficient methods for computing with matrices over finite fields often involve randomised algorithms, where matrices with a certain property are sought via repeated random selection. Complexity analyses for these algorithms require knowledge of the proportion of relevant matrices in the ambient group or algebra. We introduce a method for estimating proportions of families $N$ of elements in the algebra of all $d\times d$ matrices over a field of order $q$, where membership of a matrix in $N$ depends only on its `invertible part'. The method is based on estimating proportions of certain subsets of $\text{GL}(d,q)$ depending on $N$, so that existing estimation techniques for nonsingular matrices can be leveraged to deal with families containing singular matrices. As an application we investigate primary cyclic matrices, which are used in the Holt-Rees MEAT-AXE algorithm for testing irreducibility of matrix algebras.

math.GR↗

arXiv2012

Abundant p-singular elements in finite classical groups

In 1995, Isaacs, Kantor and Spaltenstein proved that for a finite simple classical group G defined over a field with q elements, and for a prime divisor p of |G| distinct from the characteristic, the proportion of p-singular elements in G (elements with order divisible by p) is at least a constant multiple of (1 - 1/p)/e, where e is the order of q modulo p. Motivated by algorithmic applications, we define a subfamily of p-singular elements, called p-abundant elements, which leave invariant certain "large" subspaces of the natural G-module. We find explicit upper and lower bounds for the proportion of p-abundant elements in G, and prove that it approaches a (positive) limiting value as the dimension of G tends to infinity. It turns out that the limiting proportion of p-abundant elements is at least a constant multiple of the Isaacs-Kantor-Spaltenstein lower bound for the proportion of all p-singular elements.

math.GR↗