Luke Morgan

activity

20142025

most citedElusive groups of automorphisms of digraphs of small valency

13 citations · 42 across the 22 of their papers we have counts for

collaborators

26 papers

math.GR2025

Finite permutation groups with quasi-semiregular elements

Michael Giudici, Luke Morgan, Cheryl E. Praeger

A quasi-semiregular element in a permutation group is an element that has a unique fixed point and acts semiregularly on the remaining points. Such elements were first studied in t…

math.GR2024

Prime power coverings of groups

Michael Giudici, Luke Morgan, Cheryl E. Praeger

For a finite group $A$ with normal subgroup $G$ , a subgroup $U$ of $G$ is an $A$ -prime-power-covering subgroup if $U$ meets every $A$ -conjugacy-class of elements of $G$ of prime po…

math.GR2024★ 1 cited

Finite simple groups have many classes of $p$ -elements

Michael Giudici, Luke Morgan, Cheryl E. Praeger

For an element $x$ of a finite group $T$ , the $Aut (T)$ -class of $x$ is the set ${x^{σ} ∣ σ \in Aut (T)}$ . We prove that the order $∣ T ∣$ of a finite nonabelian si…

math.CO2023

Resolution of a conjecture about linking ring structures

Marston Conder, Luke Morgan, Primož Potočnik

An LR-structure is a tetravalent vertex-transitive graph together with a special type of a decomposition of its edge-set into cycles. LR-structures were introduced in a paper by P.…

math.CO2022

A new infinite family of star normal quotient graphs of twisted wreath type

Eda Kaja, Luke Morgan

We construct the first infinite families of locally arc transitive graphs with the property that the automorphism group has two orbits on vertices and is quasiprimitive on exactly…

math.CO2021★ 1 cited

On primitive $2$ -closed permutation groups of rank at most four

Michael Giudici, Luke Morgan, Jin-Xin Zhou

We characterise the primitive 2-closed groups $G$ of rank at most four that are not the automorphism group of a graph or digraph and show that if the degree is at least 2402 then t…

math.GR2020

Vertex-transitive graphs with local action the symmetric group on ordered pairs

Luke Morgan

We consider a finite, connected and simple graph $Γ$ that admits a vertex-transitive group of automorphisms $G$ . Under the assumption that, for all $x \in V (Γ)$ , the local action $…

math.CO2019

The NNN-Property of Cyclic Groups

Michael Giudici, Luke Morgan, Yian Xu

A Cayley graph is said to be an NNN-graph if it is both normal and non-normal for isomorphic regular groups, and a group has the NNN-property if there exists an NNN-graph for it. I…

math.GR2019

Generalised shuffle groups

Carmen Amarra, Luke Morgan, Cheryl E. Praeger

The mathematics of shuffling a deck of $2 n$ cards with two "perfect shuffles" was brought into clarity by Diaconis, Graham and Kantor. Here we consider a generalisation of this pro…

math.GR2018

The distinguishing number of quasiprimitive and semiprimitive groups

Alice Devillers, Scott Harper, Luke Morgan

The distinguishing number of $G \leqslant \sym(Ω)$ is the smallest size of a partition of $Ω$ such that only the identity of $G$ fixes all the parts of the partition. Extending ear…

math.GR2018

Bounds for Finite Semiprimitive Permutation Groups: Order, Base Size, and Minimal Degree

Luke Morgan, Cheryl E. Praeger, Kyle Rosa

In this paper we study finite semiprimitive permutation groups, that is, groups in which each normal subgroup is transitive or semiregular. We give bounds on the order, base size,…

math.GR2018

Coprime Subdegrees of Twisted Wreath Permutation Groups

Alexander Y. Chua, Michael Giudici, Luke Morgan

Dolfi, Guralnick, Praeger and Spiga asked if there exist infinitely many primitive groups of twisted wreath type with nontrivial coprime subdegrees. Here we settle this question in…

math.CO2017

Arc-transitive digraphs of given out-valency and with blocks of given size

Luke Morgan, Primoz Potocnik, Gabriel Verret

Given integers $k$ and $m$ , we construct a $G$ -arc-transitive graph of valency $k$ and an $L$ -arc-transitive oriented digraph of out-valency $k$ such that $G$ and $L$ both admit bl…

math.GR2017★ 1 cited

On $p$ -groups with automorphism groups related to the Chevalley group $G_{2} (p)$

John Bamberg, Saul D. Freedman, Luke Morgan

Let $p$ be an odd prime. We construct a $p$ -group $P$ of nilpotency class two, rank seven and exponent $p$ , such that $Aut (P)$ induces $N_{\mathrm{GL}(7,p)}(G_2(p)) = Z(\m…

math.GR2017

A finite simple group is CCA if and only if it has no element of order four

Luke Morgan, Joy Morris, Gabriel Verret

A Cayley graph for a group $G$ is CCA if every automorphism of the graph that preserves the edge-orbits under the regular representation of $G$ is an element of the normaliser of $…

math.CO2017

Digraphs with small automorphism groups that are Cayley on two nonisomorphic groups

Luke Morgan, Joy Morris, Gabriel Verret

Let $Γ = Cay (G, S)$ be a Cayley digraph on a group $G$ and let $A = Aut (Γ)$ . The Cayley index of $Γ$ is $∣ A : G ∣$ . It has previously been shown that, if $p$ is a prime,…

math.CO2017

Characterising CCA Sylow cyclic groups whose order is not divisible by four

Luke Morgan, Joy Morris, Gabriel Verret

A Cayley graph on a group $G$ has a natural edge-colouring. We say that such a graph is CCA if every automorphism of the graph that preserves this edge-colouring is an element of t…

math.GR2016

A theory of semiprimitive groups

Michael Giudici, Luke Morgan

A transitive permutation group is semiprimitive if each of its normal subgroups is transitive or semiregular. Interest in this class of groups is motivated by two sources: problems…

math.GR2016

Maximal linear groups induced on the Frattini quotient of a $p$ -group

John Bamberg, S. P. Glasby, Luke Morgan +1

Let $p > 3$ be a prime. For each maximal subgroup $H ⩽ GL (d, p)$ with $∣ H ∣ ⩾ p^{3 d + 1}$ , we construct a $d$ -generator finite $p$ -group $G$ with the property that…

math.GR2015

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

Luke Morgan, Tomasz Popiel

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 a…

math.GR2015★ 4 cited

A note on the probability of generating alternating or symmetric groups

Luke Morgan, Colva M. Roney-Dougal

We improve on recent estimates for the probability of generating the alternating and symmetric groups $Alt (n)$ and $Sym (n)$ . In particular we find the sharp lower…

math.CO2014

On the order of Borel subgroups of group amalgams and an application to locally-transitive graphs

Luke Morgan, Pablo Spiga, Gabriel Verret

A permutation group is called semiprimitive if each of its normal subgroups is either transitive or semiregular. Given nontrivial finite transitive permutation groups $L_{1}$ and $L_…

math.GR2014

A characterisation of weakly locally projective amalgams related to $A_{16}$ and the sporadic simple groups $M_{24}$ and $H e$

Michael Giudici, A. A. Ivanov, Luke Morgan +1

A simple undirected graph is weakly $G$ -locally projective, for a group of automorphisms $G$ , if for each vertex $x$ , the stabiliser $G (x)$ induces on the set of vertices adjacent…

math.GR2014★ 11 cited

On locally semiprimitive graphs and a theorem of Weiss

Michael Giudici, Luke Morgan

In this paper we investigate graphs that admit a group acting arc-transitively such that the local action is semiprimitive with a regular normal nilpotent subgroup. This type of se…

Finite permutation groups with quasi-semiregular elements

Prime power coverings of groups

Finite simple groups have many classes of p-elements

Resolution of a conjecture about linking ring structures

A new infinite family of star normal quotient graphs of twisted wreath type

On primitive 2-closed permutation groups of rank at most four

Vertex-transitive graphs with local action the symmetric group on ordered pairs

The NNN-Property of Cyclic Groups

Generalised shuffle groups

The distinguishing number of quasiprimitive and semiprimitive groups

Bounds for Finite Semiprimitive Permutation Groups: Order, Base Size, and Minimal Degree

Coprime Subdegrees of Twisted Wreath Permutation Groups

Arc-transitive digraphs of given out-valency and with blocks of given size

On p-groups with automorphism groups related to the Chevalley group G2​(p)

A finite simple group is CCA if and only if it has no element of order four

Digraphs with small automorphism groups that are Cayley on two nonisomorphic groups

Characterising CCA Sylow cyclic groups whose order is not divisible by four

A theory of semiprimitive groups

Maximal linear groups induced on the Frattini quotient of a p-group

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

A note on the probability of generating alternating or symmetric groups

On the order of Borel subgroups of group amalgams and an application to locally-transitive graphs

A characterisation of weakly locally projective amalgams related to A16​ and the sporadic simple groups M24​ and He

On locally semiprimitive graphs and a theorem of Weiss

Finite simple groups have many classes of $p$ -elements

On primitive $2$ -closed permutation groups of rank at most four

On $p$ -groups with automorphism groups related to the Chevalley group $G_{2} (p)$

Maximal linear groups induced on the Frattini quotient of a $p$ -group

A characterisation of weakly locally projective amalgams related to $A_{16}$ and the sporadic simple groups $M_{24}$ and $H e$