Luke Morgan
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Luke Morgan has a total of 22 co-authors. Here are the top co-authors by number of shared publications:
- Michael Giudici
The University of Western Australia
7 shared publications - Gabriel Verret
University of Western Australia
4 shared publications - Gabriel Verret
University of Auckland
4 shared publications - Joy Morris
University of Lethbridge
4 shared publications - Primož Potočnik
Institute of Mathematics Physics and Mechanics, Ljubljana, Slovenia
4 shared publications - Cheryl E. Praeger
The University of Western Australia
3 shared publications
Latest Publications
- Open Access
- 29 September 2026
- Published by Cambridge University Press (CUP) in Forum of Mathematics, Sigma
- Vol. 14
Prime power coverings of groups and field extensions
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-power order. It is conjectured that $|G:U|$ StartAbsoluteValue upper G colon upper U EndAbsoluteValue is bounded by some function of $|A:G|$ StartAbsoluteValue upper A colon upper G EndAbsoluteValue . We prove the conjecture in the case that the action of G on the set of right cosets of U in G is innately transitive. This includes the special case where U is a maximal subgroup of G . It turns out that each prime-power-covering subgroup U for which this G -action is innately transitive must be a maximal subgroup of G . We derive the following number-theoretic consequence: if K is a finite Galois extension of a global field k and L is a finite innate extension of K such that K and L are weakly Kronecker equivalent over k , then the extension $L/K$ upper L divided by upper K must be atomic of degree bounded in terms of $[K:k]$ left bracket upper K colon k right bracket . Our proofs use both a recently proved bound on the order of a finite nonabelian simple group in terms of the number of its classes of elements of prime-power order (depending on the Finite Simple Group Classification) and also new Galois-theoretic concepts for field extensions.- 29 September 2026
- Published by Cambridge University Press (CUP) in Forum of Mathematics, Sigma
- Vol. 14
Open AccessPrime power coverings of groups and field extensions
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-power order. It is conjectured that $|G:U|$ StartAbsoluteValue upper G colon upper U EndAbsoluteValue is bounded by some function of $|A:G|$ StartAbsoluteValue upper A colon upper G EndAbsoluteValue . We prove the conjecture in the case that the action of G on the set of right cosets of U in G is innately transitive. This includes the special case where U is a maximal subgroup of G . It turns out that each prime-power-covering subgroup U for which this G -action is innately transitive must be a maximal subgroup of G . We derive the following number-theoretic consequence: if K is a finite Galois extension of a global field k and L is a finite innate extension of K such that K and L are weakly Kronecker equivalent over k , then the extension $L/K$ upper L divided by upper K must be atomic of degree bounded in terms of $[K:k]$ left bracket upper K colon k right bracket . Our proofs use both a recently proved bound on the order of a finite nonabelian simple group in terms of the number of its classes of elements of prime-power order (depending on the Finite Simple Group Classification) and also new Galois-theoretic concepts for field extensions. - Open Access
- 18 June 2026
- Published by Elsevier in Journal of Algebra
- Vol. 709, 190-220
Almost maximal permutation representations of finite simple groups
- 18 June 2026
- Published by Elsevier in Journal of Algebra
- Vol. 709, 190-220
Open AccessAlmost maximal permutation representations of finite simple groups
- Open Access
- 10 March 2026
- Published by Elsevier in Journal of Algebra
- Vol. 698, 86-115
Finite permutation groups with quasi-semiregular elements
- 10 March 2026
- Published by Elsevier in Journal of Algebra
- Vol. 698, 86-115
Open AccessFinite permutation groups with quasi-semiregular elements
- Open Access
Linear dimension of group actions
Two fundamental ways to represent a group are as permutations and as matrices. In this paper, we study linear representations of groups that intertwine with a permutation representation. Recently, D'Alconzo and Di Scala investigated how small the matrices in such a linear representation can be. The minimal dimension of such a representation is the \emph{linear dimension of the group action} and this has applications in cryptography and cryptosystems. We develop the idea of linear dimension from an algebraic point of view by using the theory of permutation modules. We give structural results about representations of minimal dimension and investigate the implications of faithfulness, transitivity and primitivity on the linear dimension. Furthermore, we compute the linear dimension of several classes of finite primitive permutation groups. We also study wreath products, allowing us to determine the linear dimension of imprimitive group actions. Finally, we give the linear dimension of almost simple finite $2$-transitive groups, some of which may be used for further applications in cryptography. Our results also open up many new questions about linear representations of group actions.
Open AccessLinear dimension of group actions
Two fundamental ways to represent a group are as permutations and as matrices. In this paper, we study linear representations of groups that intertwine with a permutation representation. Recently, D'Alconzo and Di Scala investigated how small the matrices in such a linear representation can be. The minimal dimension of such a representation is the \emph{linear dimension of the group action} and this has applications in cryptography and cryptosystems. We develop the idea of linear dimension from an algebraic point of view by using the theory of permutation modules. We give structural results about representations of minimal dimension and investigate the implications of faithfulness, transitivity and primitivity on the linear dimension. Furthermore, we compute the linear dimension of several classes of finite primitive permutation groups. We also study wreath products, allowing us to determine the linear dimension of imprimitive group actions. Finally, we give the linear dimension of almost simple finite $2$-transitive groups, some of which may be used for further applications in cryptography. Our results also open up many new questions about linear representations of group actions.
- Open Access
- 26 May 2025
- Published by Mathematical Sciences Publishers in Pacific Journal of Mathematics
- Vol. 336 (1-2) , 137-160
Finite simple groups have many classes of p-elements
For an element x of a finite group T , the Aut(T)-class of x is {x σ | σ ∈ Aut(T)}. We prove that the order |T | of a finite nonabelian simple group T is bounded above by a function of the parameter m(T), where m(T) is the maximum, over all primes p, of the number of Aut(T)-classes of elements of T of p-power order. This bound is a substantial generalisation of the results of Pyber (1992) and of Héthelyi and Külshammer (2005), and it has implications for relative Brauer groups of finite extensions of global fields.
- 26 May 2025
- Published by Mathematical Sciences Publishers in Pacific Journal of Mathematics
- Vol. 336 (1-2) , 137-160
Open AccessFinite simple groups have many classes of p-elements
For an element x of a finite group T , the Aut(T)-class of x is {x σ | σ ∈ Aut(T)}. We prove that the order |T | of a finite nonabelian simple group T is bounded above by a function of the parameter m(T), where m(T) is the maximum, over all primes p, of the number of Aut(T)-classes of elements of T of p-power order. This bound is a substantial generalisation of the results of Pyber (1992) and of Héthelyi and Külshammer (2005), and it has implications for relative Brauer groups of finite extensions of global fields.
- 25 May 2023
- Published by Elsevier in Journal of Algebra
- Vol. 632, 87-101
Resolution of a conjecture about linking ring structures
- 25 May 2023
- Published by Elsevier in Journal of Algebra
- Vol. 632, 87-101
Resolution of a conjecture about linking ring structures
- Open Access
Resolution of a conjecture about linking ring structures
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. Potočnik and S. Wilson, titled `Linking rings structures and tetravalent semisymmetric graphs', in Ars Math. Contemp. 7 (2014), as a tool to study tetravalent semisymmetric graphs of girth 4. In this paper, we use the methods of group amalgams to resolve some problems left open in the above-mentioned paper.
Open AccessResolution of a conjecture about linking ring structures
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. Potočnik and S. Wilson, titled `Linking rings structures and tetravalent semisymmetric graphs', in Ars Math. Contemp. 7 (2014), as a tool to study tetravalent semisymmetric graphs of girth 4. In this paper, we use the methods of group amalgams to resolve some problems left open in the above-mentioned paper.
- Open Access
Bounds for Finite Semiprimitive Permutation Groups: Order, Base Size, and Minimal Degree
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, minimal degree, fixity, and chief length of an arbitrary finite semiprimitive group in terms of its degree. To establish these bounds, we classify finite semiprimitive groups that induce the alternating or symmetric group on the set of orbits of an intransitive normal subgroup.
Open AccessBounds for Finite Semiprimitive Permutation Groups: Order, Base Size, and Minimal Degree
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, minimal degree, fixity, and chief length of an arbitrary finite semiprimitive group in terms of its degree. To establish these bounds, we classify finite semiprimitive groups that induce the alternating or symmetric group on the set of orbits of an intransitive normal subgroup.
- Open Access
Coprime Subdegrees of Twisted Wreath Permutation Groups
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 the affirmative. We construct infinite families of primitive twisted wreath permutation groups with nontrivial coprime subdegrees. In particular, we define a primitive twisted wreath group $G(m,q)$ constructed from the nonabelian simple group $\text{PSL}(2,q)$ and a primitive permutation group of diagonal type with socle $\text{PSL}(2,q)^m$, and determine all values of $m$ and $q$ for which $G(m,q)$ has nontrivial coprime subdegrees. In the case where $m=2$ and $q\notin\{7,11,29\}$ we obtain a full classification of all pairs of nontrivial coprime subdegrees.Open AccessCoprime Subdegrees of Twisted Wreath Permutation Groups
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 the affirmative. We construct infinite families of primitive twisted wreath permutation groups with nontrivial coprime subdegrees. In particular, we define a primitive twisted wreath group $G(m,q)$ constructed from the nonabelian simple group $\text{PSL}(2,q)$ and a primitive permutation group of diagonal type with socle $\text{PSL}(2,q)^m$, and determine all values of $m$ and $q$ for which $G(m,q)$ has nontrivial coprime subdegrees. In the case where $m=2$ and $q\notin\{7,11,29\}$ we obtain a full classification of all pairs of nontrivial coprime subdegrees. - Open Access
Arc-transitive digraphs of given out-valency and with blocks of given size
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 blocks of imprimitivity of size $m$.Open AccessArc-transitive digraphs of given out-valency and with blocks of given size
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 blocks of imprimitivity of size $m$.