S. Glasby

activity

20032025

most citedModules induced from a normal subgroup of prime index

4 citations · 19 across the 29 of their papers we have counts for

collaborators

34 papers

math.NT2025

Hilbert's Theorem 90, periodicity, and roots of Artin-Schreier polynomials

S. P. Glasby

Let $E / F$ be a cyclic field extension of degree $n$ , and let $σ$ generate the group $Gal (E / F)$ . If $Tr_{F}^{E} (y) = \sum_{i = 0}^{n - 1} σ^{i} y = 0$ , then the additive form of Hilber…

math.RT2024

Absolutely irreducible quasisimple linear groups containing elements of order a specified Zsigmondy prime

S. P. Glasby, Alice C. Niemeyer, Cheryl E. Praeger +1

This paper is concerned with absolutely irreducible quasisimple subgroups $G$ of a finite general linear group $G L_{d} (F_{q})$ for which some element $g \in G$ of prime order $…

math.GR2024

Classifying finite groups G with three Aut(G)-orbits

Stephen P. Glasby

We give a complete and irredundant list of the finite groups $G$ for which Aut $(G)$ , acting naturally on $G$ , has precisely $3$ orbits. There are 7 infinite families: one abelian,…

math.GR2023

Bipartite $q$ -Kneser graphs and two-generated irreducible linear groups

S. P. Glasby, Alice C. Niemeyer, Cheryl E. Praeger

Let $V := (F_{q})^{d}$ be a $d$ -dimensional vector space over the field $F_{q}$ of order $q$ . Fix positive integers $e_{1}, e_{2}$ satisfying $e_{1} + e_{2} = d$ . Motivated by analysi…

math.NT2023

Maximizing weighted sums of binomial coefficients using generalized continued fractions

S. P. Glasby, G. R. Paseman

Let $m, r \in Z$ and $ω \in R$ satisfy $0 ⩽ r ⩽ m$ and $ω ⩾ 1$ . Our main result is a generalized continued fraction for an expression involving the…

math.GR2022

Derangements in wreath products of permutation groups

Vishnuram Arumugam, Heiko Dietrich, S. P. Glasby

Given a finite group $G$ acting on a set $X$ let $δ_{k} (G, X)$ denote the proportion of elements in $G$ that have exactly $k$ fixed points in $X$ . Let $S_{n}$ denote the symmet…

math.CO2022★ 1 cited

Tournaments and Even Graphs are Equinumerous

Gordon F. Royle, Cheryl E. Praeger, S. P. Glasby +2

A graph is called odd if there is an orientation of its edges and an automorphism that reverses the sense of an odd number of its edges, and even otherwise. Pontus von Brömssen (né…

math.GR2021

Random generation of direct sums of finite non-degenerate subspaces

S. P. Glasby, Alice C. Niemeyer, Cheryl E. Praeger

Let $V$ be a $d$ -dimensional vector space over a finite field $F$ equipped with a non-degenerate hermitian, alternating, or quadratic form. Suppose $∣ F ∣ = q^{2}$ if $…

math.GR2021

The groups $G$ satisfying a functional equation $f (x k) = x f (x)$ for some $k \in G$

Dominik Bernhardt, Tim Boykett, Alice Devillers +2

We study the groups $G$ with the curious property that there exists an element $k \in G$ and a function $f : G \to G$ such that $f (x k) = x f (x)$ holds for all $x \in G$ . This propert…

math.RT2021★ 4 cited

Modules induced from a normal subgroup of prime index

S. P. Glasby

Let $G$ be a finite group and $H$ a normal subgroup of prime index $p$ . Let $V$ be an irreducible $F H$ -module and $U$ a quotient of the induced $F G$ -module $V\k…

math.GR2020★ 1 cited

Subgroups of Classical Groups that are Transitive on Subspaces

Michael Giudici, S. P. Glasby, Cheryl E. Praeger

For each finite classical group $G$ , we classify the subgroups of $G$ which act transitively on a $G$ -invariant set of subspaces of the natural module, where the subspaces are eith…

math.GR2020

Point-primitive generalised hexagons and octagons and projective linear groups

S. P. Glasby, E. Pierro, Cheryl E. Praeger

We discuss recent progress on the problem of classifying point-primitive generalised polygons. In the case of generalised hexagons and generalised octagons, this has reduced the pr…

math.GR2020

Finite $2$ -groups with exactly three automorphism orbits

Alexander Bors, Stephen P. Glasby

We give a complete classification of the finite $2$ -groups $G$ for which the automorphism group $Aut (G)$ acting naturally on $G$ has three orbits. There are two infi…

math.GR2020

Conjugacy class sizes in arithmetic progression

Mariagrazia Bianchi, Cheryl E. Praeger, S. P. Glasby

Let $cs (G)$ denote the set of conjugacy class sizes of a group $G$ , and let $cs^{*} (G) = cs (G) ∖ {1}$ be the sizes of non-central classes. We prove three resu…

math.NT2019

Cyclotomic ordering conjecture

S. P. Glasby

This note describes a conjecture involving cyclotomic polynomials and some initial thoughts towards a solution. Given positive integers $m, n$ , the conjecture is that either $Φ_m(q)…

math.GR2019

Classifying uniformly generated groups

S. P. Glasby

A finite group $G$ is called *uniformly generated*, if whenever there is a (strictly ascending) chain of subgroups $1<\langle x_1\rangle<\langle x_1,x_2\rangle <\cdots<\langle x_1,…

math.NT2019★ 1 cited

Roots of unity and unreasonable differentiation

Alice Devillers, S. P. Glasby

We explore when it is legal to differentiate a polynomial evaluated at a root of unity using modular arithmetic.

math.GR2018

Involution centralisers in finite unitary groups of odd characteristic

S. P. Glasby, Cheryl E. Praeger, Colva M. Roney-Dougal

We analyse the complexity of constructing involution centralisers in unitary groups over fields of odd order. In particular, we prove logarithmic bounds on the number of random ele…

math.CO2018

Permutations with orders coprime to a given integer

John Bamberg, S. P. Glasby, Scott Harper +1

Let $m$ be a positive integer and let $ρ (m, n)$ be the proportion of permutations of the symmetric group $Sym (n)$ whose order is coprime to $m$ . In 2002, Pouyanne proved that…

math.GR2018★ 2 cited

The shape of solvable groups with odd order

S. P. Glasby

We prove that the minimal composition length, $c$ , of a solvable group with solvable length $d$ satisfies $9^{(d - 3) /9} < c < 9^{(d + 1) /5}$ , and the minimal composition length, $c^{o}$ ,…

math.GR2017★ 1 cited

Bounding the composition length of primitive permutation groups and completely reducible linear groups

S. P. Glasby, Cheryl E. Praeger, Kyle Rosa +1

We obtain upper bounds on the composition length of a finite permutation group in terms of the degree and the number of orbits, and analogous bounds for primitive, quasiprimitive a…

math.CO2017★ 1 cited

On the parameters of intertwining codes

S. P. Glasby, Cheryl E. Praeger

Let $F$ be a field and let $F^{r \times s}$ denote the space of $r \times s$ matrices over $F$ . Given equinumerous subsets $A = {A_{i} ∣ i \in I} \subseteq F^{r \times r}$ an…

math.CO2016

Twisted Centralizer Codes

Adel Alahmadi, S. P. Glasby, Cheryl E. Praeger +2

Given an $n \times n$ matrix $A$ over a field $F$ and a scalar $a \in F$ , we consider the linear codes $C (A, a) := {B \in F^{n \times n} ∣ A B = a B A}$ of length $n^{2}$ . We call $C (A, a)$ …

math.CO2016★ 2 cited

On the dimension of twisted centralizer codes

S. P. Glasby, Cheryl E. Praeger, Adel Alahmadi

Given a field $F$ , a scalar $λ \in F$ and a matrix $A \in F^{n \times n}$ , the twisted centralizer code $C_{F} (A, λ) := {B \in F^{n \times n} ∣ A B - λ B A = 0}$ is a linear code of length $n^2…

Hilbert's Theorem 90, periodicity, and roots of Artin-Schreier polynomials

Absolutely irreducible quasisimple linear groups containing elements of order a specified Zsigmondy prime

Classifying finite groups G with three Aut(G)-orbits

Bipartite q-Kneser graphs and two-generated irreducible linear groups

Maximizing weighted sums of binomial coefficients using generalized continued fractions

Derangements in wreath products of permutation groups

Tournaments and Even Graphs are Equinumerous

Random generation of direct sums of finite non-degenerate subspaces

The groups G satisfying a functional equation f(xk)=xf(x) for some k∈G

Modules induced from a normal subgroup of prime index

Subgroups of Classical Groups that are Transitive on Subspaces

Point-primitive generalised hexagons and octagons and projective linear groups

Finite 2-groups with exactly three automorphism orbits

Conjugacy class sizes in arithmetic progression

Cyclotomic ordering conjecture

Classifying uniformly generated groups

Roots of unity and unreasonable differentiation

Involution centralisers in finite unitary groups of odd characteristic

Permutations with orders coprime to a given integer

The shape of solvable groups with odd order

Bounding the composition length of primitive permutation groups and completely reducible linear groups

On the parameters of intertwining codes

Twisted Centralizer Codes

On the dimension of twisted centralizer codes

Bipartite $q$ -Kneser graphs and two-generated irreducible linear groups

The groups $G$ satisfying a functional equation $f (x k) = x f (x)$ for some $k \in G$

Finite $2$ -groups with exactly three automorphism orbits