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Publications (25)

math.GR2020

The Spread of Almost Simple Classical Groups

Scott Harper

Every finite simple group can be generated by two elements, and in 2000, Guralnick and Kantor resolved a 1962 question of Steinberg by proving that in a finite simple group every n…

cs.RO2021

NeBula: Quest for Robotic Autonomy in Challenging Environments; TEAM CoSTAR at the DARPA Subterranean Challenge

Ali Agha, Kyohei Otsu, Benjamin Morrell +69

This paper presents and discusses algorithms, hardware, and software architecture developed by the TEAM CoSTAR (Collaborative SubTerranean Autonomous Robots), competing in the DARP…

math.GR2023

The maximal size of a minimal generating set

Scott Harper

A generating set for a finite group $G$ is said to be minimal if no proper subset generates $G$ , and $m (G)$ denotes the maximal size of a minimal generating set for $G$ . We prove a…

math.GR2018

On the uniform domination number of a finite simple group

Timothy C. Burness, Scott Harper

Let $G$ be a finite simple group. By a theorem of Guralnick and Kantor, $G$ contains a conjugacy class $C$ such that for each non-identity element $x \in G$ , there exists $y \in C$ …

math.GR2026

The probability of generating finite and profinite groups

Scott Harper, Martyn Quick

Famously, every finite simple group $G$ can be generated by a pair of elements. Moreover, Liebeck and Shalev (1995) proved that the probability that a pair of elements generate $G$ …

math.GR2024

Representations of extensions of simple groups

Scott Harper, Martin W. Liebeck

Feit and Tits (1978) proved that a nontrivial projective representation of minimal dimension of a finite extension of a finite nonabelian simple group $G$ factors through a project…

math.GR2024

Minimal cover groups

Peter J. Cameron, David Craven, Hamid Reza Dorbidi +2

Let $F$ be a set of finite groups. A finite group $G$ is called an \emph{ $F$ -cover} if every group in $F$ is isomorphic to a subgroup of $G$ . An $\mat…

math.GR2024

Totally deranged elements of almost simple groups and invariable generating sets

Scott Harper

By a classical theorem of Jordan, every faithful transitive action of a nontrivial finite group has a derangement (an element with no fixed points). The existence of derangements w…

math.GR2023

The spread of finite and infinite groups

Scott Harper

It is well known that every finite simple group has a generating pair. Moreover, Guralnick and Kantor proved that every finite simple group has the stronger property, known as $\fr…

math.GR2020

Connectivity of generating graphs of nilpotent groups

Scott Harper, Andrea Lucchini

Let $G$ be $2$ -generated group. The generating graph of $I^ (G)$ is the graph whose vertices are the elements of $G$ and where two vertices $g$ and $h$ are adjacent if $G=\langle g,…

math.CO2019

Permutations with orders coprime to a given integer

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

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

cs.RO2018

Design of an Autonomous Precision Pollination Robot

Nicholas Ohi, Kyle Lassak, Ryan Watson +17

Precision robotic pollination systems can not only fill the gap of declining natural pollinators, but can also surpass them in efficiency and uniformity, helping to feed the fast-g…

math.GR2026

Derangements in intransitive groups

David Ellis, Scott Harper

Let $G$ be a nontrivial permutation group of degree $n$ . If $G$ is transitive, then a theorem of Jordan states that $G$ has a derangement. Equivalently, a finite group is never the…

math.GR2026

Generating simple vigorous groups

Collin Bleak, Casey Donoven, Scott Harper +1

The simple vigorous groups form a broad class of groups of homeomorphisms of Cantor space that includes Thompson's group $V$ , its various generalisations and many others such as Ne…

math.GR2021

The spread of a finite group

Timothy C. Burness, Robert M. Guralnick, Scott Harper

A group $G$ is said to be $\frac{3}{2}$ -generated if every nontrivial element belongs to a generating pair. It is easy to see that if $G$ has this property then every proper quotie…

cs.RO2019

Improved Planetary Rover Inertial Navigation and Wheel Odometry Performance through Periodic Use of Zero-Type Constraints

Cagri Kilic, Jason N. Gross, Nicholas Ohi +5

We present an approach to enhance wheeled planetary rover dead-reckoning localization performance by leveraging the use of zero-type constraint equations in the navigation filter.…

math.GR2021

Shintani descent, simple groups and spread

Scott Harper

The spread of a group $G$ , written $s (G)$ , is the largest $k$ such that for any nontrivial elements $x_{1}, \dots, x_{k} \in G$ there exists $y \in G$ such that $G = \langle x_i, y \ra…

math.GR2019

The distinguishing number of quasiprimitive and semiprimitive groups

Alice Devillers, Scott Harper, Luke Morgan

The distinguishing number of $G \leqslant \sym(Î©)$ is the smallest size of a partition of $I^c ◯$ such that only the identity of $G$ fixes all the parts of the partition. Extending e…

math.GR2017

On the uniform spread of almost simple symplectic and orthogonal groups

Scott Harper

A group is $\frac{3}{2}$ -generated if every non-identity element is contained in a generating pair. A conjecture of Breuer, Guralnick and Kantor from 2008 asserts that a finite gro…

cs.LO2025

Classifying the groups of order $pq$ in Lean

Scott Harper, Peiran Wu

This note discusses our formalisation in Lean of the classification of the groups of order $pq$ for (not necessarily distinct) prime numbers $p$ and $q$ , together with various int…

math.GR2024

Kronecker classes, normal coverings and chief factors of groups

Marco Fusari, Scott Harper, Pablo Spiga

For a group $G$ , a subgroup $U \leq G$ and a group $Inn (G) \leq A \leq Aut (G)$ , we say that $U$ is an $A$ -covering group of $G$ if $G = ⋃_{a \in A} U^{a}$ . A th…

math.GR2023

Thompson's group $T$ is $\frac{3}{2}$ -generated

Collin Bleak, Scott Harper, Rachel Skipper

Every finite simple group can be generated by two elements and, in fact, every nontrivial element is contained in a generating pair. Groups with this property are said to be $\frac…

math.GR2021

Flexibility in generating sets of finite groups

Scott Harper

Let G be a finite group. It has recently been proved that every nontrivial element of G is contained in a generating set of minimal size if and only if all proper quotients of G re…

math.GR2020

Infinite $\frac{3}{2}$ -generated groups

Casey Donoven, Scott Harper

Every finite simple group can be generated by two elements, and Guralnick and Kantor proved that, moreover, every nontrivial element is contained in a generating pair. Groups with…

math.GR2019

Finite groups, 2-generation and the uniform domination number

Timothy C. Burness, Scott Harper

Let $G$ be a finite $2$ -generated non-cyclic group. The spread of $G$ is the largest integer $k$ such that for any nontrivial elements $x_{1}, \dots, x_{k}$ , there exists $y \in G$ su…

Publications (25)

The Spread of Almost Simple Classical Groups

NeBula: Quest for Robotic Autonomy in Challenging Environments; TEAM CoSTAR at the DARPA Subterranean Challenge

The maximal size of a minimal generating set

On the uniform domination number of a finite simple group

The probability of generating finite and profinite groups

Representations of extensions of simple groups

Minimal cover groups

Totally deranged elements of almost simple groups and invariable generating sets

The spread of finite and infinite groups

Connectivity of generating graphs of nilpotent groups

Permutations with orders coprime to a given integer

Design of an Autonomous Precision Pollination Robot

Derangements in intransitive groups

Generating simple vigorous groups

The spread of a finite group

Improved Planetary Rover Inertial Navigation and Wheel Odometry Performance through Periodic Use of Zero-Type Constraints

Shintani descent, simple groups and spread

The distinguishing number of quasiprimitive and semiprimitive groups

On the uniform spread of almost simple symplectic and orthogonal groups

Classifying the groups of order pq in Lean

Kronecker classes, normal coverings and chief factors of groups

Thompson's group T is 23​-generated

Flexibility in generating sets of finite groups

Infinite 23​-generated groups

Finite groups, 2-generation and the uniform domination number

Classifying the groups of order $pq$ in Lean

Thompson's group $T$ is $\frac{3}{2}$ -generated

Infinite $\frac{3}{2}$ -generated groups