## About

76

Publications

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Introduction

My main research interests lie in the area of Computational Group Theory, Statistical
Group Theory, Combinatorics and Simplicial Surfaces.

Additional affiliations

October 1993 - December 2014

## Publications

Publications (76)

Nature, one of the biggest sources of inspiration for civil engineering design, exhibits thin-walled shell structures among a wide range of organisms. Due to membrane stresses that ideally prevail in such thin-walled structures, shells have an extremely favourable span-to-material ratio that in turn means excellent structural performance. The appro...

The art of origami proved to be a remarkably useful technique for technical applications in many fields such as architecture, engineering, medicine and aerospace. Even though existing numerical methods of rigid-origami can be used to describe the geometrical forms, algebraic solutions of the waterbomb folding kinematics and tessellations reduce the...

Let n,n′ be positive integers and let V be an (n+n′)-dimensional vector space over a finite field F equipped with a non-degenerate alternating, hermitian or quadratic form. We estimate the proportion of pairs (U,U′), where U is a non-degenerate n-subspace and U′ is a non-degenerate n′-subspace of V, such that U+U′=V (usually such spaces U and U′ ar...

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|=q2 if V is hermitian, and |F|=q otherwise. Given integers e,e′ such that e+e′⩽d, we estimate the proportion of pairs (U,U′), where U is a non-degenerate e-subspace of V and U′ is a non-degenerate e′-subsp...

In analogy to the disjoint cycle decomposition in permutation groups, Ore and Specht define a decomposition of elements of the full monomial group and exploit this to describe conjugacy classes and centralisers of elements in the full monomial group. We generalise their results to wreath products whose base group need not be finite and whose top gr...

Let $V$ be a $d$-dimensional vector space over a finite field $\mathbb{F}$ equipped with a non-degenerate hermitian, alternating, or quadratic form. Suppose $|\mathbb{F}|=q^2$ if $V$ is hermitian, and $|\mathbb{F}|=q$ otherwise. Given integers $e, e'$ such that $e+e'\leqslant d$, we estimate the proportion of pairs $(U, U')$, where $U$ is a non-deg...

Let $n,n'$ be positive integers and let $V$ be an $(n+n')$-dimensional vector space over a finite field $\mathbb{F}$ equipped with a non-degenerate alternating, hermitian or quadratic form. We estimate the proportion of pairs $(U, U')$, where $U$ is a non-degenerate $n$-subspace and $U'$ is a non-degenerate $n'$-subspace of $V$, such that $U+ U'=V$...

We present an exact algebraic solution to model waterbomb tessellations. Given the design parameters of a single waterbomb cell we can determine the global geometry of the tessellated shell directly. Thereby the preliminary design of thin-walled folded shells with load carrying function can be streamlined. The primary focus of the paper is on symme...

In analogy to the disjoint cycle decomposition in permutation groups, Ore and Specht define a decomposition of elements of the full monomial group and exploit this to describe conjugacy classes and centralisers of elements in the full monomial group. We generalise their results to wreath products whose base group need not be finite and whose top gr...

We construct a family of hemisystems of the parabolic quadric Q(2d,q), for all ranks d⩾2 and all odd prime powers q, that admit Ω3(q)≅PSL2(q). This yields the first known construction for d⩾4.

We constuct a family of hemisystems of the parabolic quadric $\mathcal{Q}(2d, q)$, for all ranks $d \ge 2$ and all odd prime powers $q$, that admit $\Omega_3(q) \cong \mathrm{PSL}_2(q)$. This yields the first known construction for $d \ge 4$.

Retaining the combinatorial Euclidean
structure of a regular icosahedron, namely the 20 equiangular
(planar) triangles, the 30 edges of length 1, and the 12 different vertices together with the incidence structure, we
investigate variations of the regular icosahedron admitting
self-intersections of faces. We determine all rigid equivalence classes...

We give a correction to Fig. 1 and supporting text published in the paper: ‘Maximal linear groups induced on the Frattini quotient of a p-group’, J. Pure Appl. Algebra 222 (10) (2018) 2931–2951.

Retaining the combinatorial Euclidean structure of a regular icosahedron, namely the 20 equiangular (planar) triangles, the 30 edges of length 1, and the 12 different vertices together with the incidence structure, we investigate variations of the regular icosahedron admitting self-intersections of faces. We determine all rigid equivalence classes...

In this paper we describe a parallel Gaussian elimination algorithm for matrices with entries in a finite field. Unlike previous approaches, our algorithm subdivides a very large input matrix into smaller submatrices by subdividing both rows and columns into roughly square blocks sized so that computing with individual blocks on individual processo...

Let $T$ be a finite simple group of Lie type in characteristic $p$, and let $S$ be a Sylow subgroup of $T$ with maximal order. It is well known that $S$ is a Sylow $p$-subgroup except in an explicit list of exceptions, and that $S$ is always `large' in the sense that $|T|^{1/3} < |S| \leqslant |T|^{1/2}$. One might anticipate that, moreover, the Sy...

Embeddings of combinatorial closed simplicial surfaces in Euclidean 3-space with all triangles congruent to one control triangle are investigated, where the control triangle may vary. Definitions and general methods for construction and classification are outlined. For one infinite family of combinatorial surfaces its dihedral symmetry is used to c...

The Divisibility Graph of a finite group $G$ has vertex set the set of conjugacy class lengths of non-central elements in $G$ and two vertices are connected by an edge if one divides the other. We determine the connected components of the Divisibility Graph of the finite groups of Lie type in odd characteristic.

Let $p>3$ be a prime. For each maximal subgroup $H \leqslant \mathrm{GL}(d,p)$ with $|H|=p^{\mathrm O(d^2)}$, we construct a $d$-generator finite $p$-group $G$ with the property that $\mathrm{Aut}(G)$ induces $H$ on the Frattini quotient $G/\Phi(G)$ and $|G|= p^{\mathrm O(d^4)}$. A significant feature of this construction is that $|G|$ is very smal...

We present new efficient data structures for elements of Coxeter groups of type and their associated Iwahori–Hecke algebras . Usually, elements of are represented as simple coefficient list of length with respect to the standard basis, indexed by the elements of the Coxeter group. In the new data structure, elements of are represented as nested coe...

Primitive prime divisors play an important role in group theory and number theory. We study a certain number theoretic quantity, called Φ * n (q), which is closely related to the cyclotomic polynomial Φ n (x) and to primitive prime divisors of q n − 1. Our definition of Φ * n (q) is novel, and we prove it is equivalent to the definition given by He...

We show that the proportion of permutations $g$ in $S_n$ or $A_n$ such that
$g$ has even order and $g^{|g|/2}$ is an involution with support of cardinality
at most $\lceil n^\varepsilon \rceil$ is at least a constant multiple of
$\varepsilon$. Using this result, we obtain the same conclusion for elements in
a classical group of natural dimension $n...

Essential steps in constructive black-box recognition algorithms for finite symmetric groups S n and alternating groups A n are the construction of n-cycles, or (n − 1)-cycles for A n with n even, and the construction of a 2-cycle or 3-cycle. The latter elements are often constructed from elements containing an m-cycle for an appropriate m ≥ n − 6.

Primitive prime divisors play an important role in group theory and number
theory. We study a certain number theoretic quantity, called $\Phi^*_n(q)$,
which is closely related to the cyclotomic polynomial $\Phi_n(x)$ and to
primitive prime divisors of $q^n-1$. Our definition of $\Phi^*_n(q)$ is novel,
and we prove it is equivalent to the definition...

We estimate the proportion of several classes of elements in finite classical
groups which are readily recognised algorithmically, and for which some power
has a large fixed point subspace and acts irreducibly on a complement of it.
The estimates are used in complexity analyses of new recognition algorithms for
finite classical groups in arbitrary...

We determine a reasonable upper bound for the complexity of collection from
the left to multiply two elements of a finite soluble, or polycyclic, group by
restricting attention to certain polycyclic presentations of the group.

This chapter discusses the role of estimation in the design and analysis of randomised algorithms for computing with finite groups.An exposition is given of a variety of different approaches to estimating proportions of important element classes, including geometric methods, and the use of generating functions and the theory of Lie type groups.Nume...

We present a constructive recognition algorithm to decide whether a given
black-box group is isomorphic to an alternating or a symmetric group without
prior knowledge of the degree. This eliminates the major gap in known
algorithms, as they require the degree as additional input.
Our methods are probabilistic and rely on results about proportions o...

We advocate that straight-line programs designed for algebraic computations
should be accompanied by a comprehensive complexity analysis that takes into
account both the number of fundamental algebraic operations needed, as well as
memory requirements arising during evaluation. We introduce an approach for
formalising this idea and, as illustration...

Probabilistic Group Theory, Combinatorics, and Computing is based on lecture courses held at the
Fifth de Brún Workshop in Galway, Ireland in April 2011. Each course discusses computational and algorithmic aspects that have recently emerged at the interface of group theory and combinatorics, with a strong focus on probabilistic methods and results...

Let G be a finite d-dimensional classical group and p a prime divisor of ∣G∣ distinct from the characteristic of the natural representation. We consider a subfamily of p-singular elements in G (elements with order divisible by p) that leave invariant a subspace X of the natural G-module of dimension greater than d/2 and either act irreducibly on X...

Let $H$ be a permutation group on a set $\Lambda$, which is permutationally
isomorphic to a finite alternating or symmetric group $A_n$ or $S_n$ acting on
the $k$-element subsets of points from $\{1,\ldots,n\}$, for some arbitrary but
fixed $k$. Suppose moreover that no isomorphism with this action is known. We
show that key elements of $H$ needed...

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...

We call an element of a finite general linear group GL(d, q) fat if it leaves invariant and acts irreducibly on a subspace of dimension greater than d/2. Fatness of an element can be decided efficiently in practice by testing whether its characteristic polynomial has an irreducible factor of degree greater than d/2. We show that for groups G with S...

In this paper we establish upper and lower bounds for the proportion of permutations in symmetric groups which power up to semiregular permutations (permutations all of whose cycles have the same length). Provided that an integer n has a divisor at most d, we show that the proportion of such elements in S n is at least cn -1+1/2d for some constant...

Key to a computational study of the finite classical groups in odd characteristic are efficient methods for constructing involutions and their centralisers. Constructing an involution in a given conjugacy class is usually achieved by finding an element of even order that powers up to an involution in the class. Lower bounds on the proportions of su...

We show that certain closure properties on subsets Q of a finite group G of Lie type enable the ratio |Q|/|G| to be determined by finding the proportions of elements of Q in the maximal tori of G and the proportions of certain related subsets of the Weyl group. We prove fundamental results about these subsets Q, including those necessary for moving...

The paper summarises existing theory and classifications for finite line-transitive linear spaces, develops the theory further,
and organises it in a way that enables its effective application. The starting point is a theorem of Camina and the fifth
author that identifies three kinds of line-transitive automorphism groups of linear spaces. In two o...

Let G be a finite group of Lie type in odd characteristic defined over a field with q elements. We prove that there is an absolute (and explicit) constant c such that, if G is a classical matrix group of dimension n⩾2, then at least c/log(n) of its elements are such that some power is an involution with fixed point subspace of dimension in the inte...

Most democratic countries use election methods to transform election results
into whole numbers which usually give the number of seats in a legislative body
the parties obtained. Which election method does this best can be specified by
measuring the error between the allocated result and the ideal proportion. We
show how to find an election method...

We study permutations of a set of size n for which the order is a multiple of n. We prove that, for large n, most such elements lie in one of two families. The first family consists of those permutations with a single very large
cycle of order dividing n and includes the n-cycles, and the second consists of permutations for which the cycles of leng...

We give a detailed analysis of the proportion of elements in the symmetric group on n points whose order divides m, for n sufficiently large and m≥n with m=O(n).

A base for a permutation group, G, is a sequence of elements of its permutation domain whose stabiliser in G is trivial. Using purely elementary and constructive methods, we obtain bounds on the minimum length of a base for the action of the symmetric group on partitions of a set into blocks of equal size. This upper bound is a constant when the si...

Burnside asked questions about periodic groups in his influential paper of 1902. The study of groups with exponent six is a special case of the study of the Burnside questions on which there has been significant progress. It has contributed a number of worthwhile aspects to the theory of groups and in particular to computation related to groups. Fi...

In 1991, Weidong Fang and Huiling Li proved that there are only finitely many
non-trivial linear spaces that admit a line-transitive, point-imprimitive group
action, for a given value of gcd(k,r), where k is the line size and r is the
number of lines on a point. The aim of this paper is to make that result
effective. We obtain a classification of a...

A general explicit upper bound is obtained for the proportion P(n,m) of elements of order dividing m, where n−1⩽m⩽cn for some constant c, in the finite symmetric group Sn. This is used to find lower bounds for the conditional probabilities that an element of Sn or An contains an r-cycle, given that it satisfies an equation of the form xrs=1 where s...

The length of every pair {A, B} of n × n complex matrices is at most 2n - 2, if n ≤ 5. That is, for n ≤ 5, the (possibly empty) words in A, B of length at most 2n - 2 span the unital algebra A generated by A, B. For every positive integer m there exist m × m complex matrices C, D such that the length of the pair {C, D} is 2m - 2.

We describe an algorithm which, for any given group G containing an ab- solutely irreducible, extraspecial normal subgroup, constructs a homomorphism, with nontrivial kernel, from G onto a nontrivial group of permutations or matrices. Thus we reduce the problem of computing with G to two smaller problems. The algorithm, which uses a blend of geomet...

We present a nearly linear-time Las Vegas algorithm that, given a large-base primitive permutation group, constructs its natural imprimitive representation.A large-base primitive permutation group is a subgroup of a wreath product of symmetric groups Sn and Sr in product action on r-tuples of k-element subsets of {1 ,...,n }, containing Ar n. The a...

We present a Las Vegas algorithm which, for a given matrix group known to be isomorphic modulo scalars to a finite alternating or symmetric group acting on the fully deleted permutation module, produces an explicit isomorphism with the standard permutation representation of the group. This algorithm exploits information available from the matrix re...

We present an algorithm that constructs and classifies finite linear spaces admitting a line-transitive group which leaves invariant a nontrivial point-partition. The algorithm was developed from an algorithm of Nickel and Niemeyer that classified such linear spaces on 729 points with line size 8, but was never published. It has been applied to com...

The members of the class C6 in Aschbacher's structure theorem for subgroups of the general linear group GL(d,q) are the normalizers of certain absolutely irreducible, symplectic-type r-groups, where r is a prime, d a power of r and q ≡ 1 (mod r). For a prime r > 2 and d = r, we present a constructive one-sided Monte Carlo algorithm to recognize whe...

We present an algorithm that constructs and classifies finite linear spaces admitting a given line-transitive group which leaves invariant a nontrivial point-partition. This algorithm was developed from an algorithm of Nickel and Niemeyer that classified such linear spaces on 729 points with line-size 8, but was never published. It has been applied...

Until the 1990's the only known finite linear spaces admitting line-transitive, point- imprimitive groups of automorphisms were Desarguesian projective planes and two linear spaces with 91 points and line size 6. In 1992 a new family of 467 such spaces was constructed, all having 729 points and line size 8. These were shown to be the only linear sp...

We present a Las Vegas algorithm which, for a given black-box group known to be isomorphic to a symmetric or alternating group, produces an explicit isomorphism with the standard permutation representation of the group. This algorithm has applications in computations with matrix groups and permutation groups.
In this paper, we handle the case when...

We present a Las Vegas algorithm which, for a given black-box group known to be isomorphic to a symmetric or alternating group, produces an explicit isomorphism with the standard permutation representation of the group. This algorithm has applications in computations with matrix groups and permutation groups. In this paper, we handle the case when...

Let G be a finite primitive permutation group with a non-trivial, non-regular normal subgroup N, and let G be an orbit of a point stabilizer Na. Then each composition factor S of Na occurs as a section of the permutation group induced by Na on G. The case N ¼ G is a theorem of Wielandt. The general result and some of its corollaries are useful for...

Short descriptions of computer algebra systems are presented in three sections: major systems, special purpose systems, and packages. However, the separation between special purpose systems and packages is not to be taken too literally. An older survey is the paper by Calmet and van Hulzen in [Buchberger et al. 1982]. There is now an excellent new...

Let q be an integer with q ≥ 2. We give a new proof of a
result of Erdös and Turán determining the proportion of
elements of the finite symmetric group Sn
having no cycle of length a multiple of q. We then extend
our methods to the more difficult case of obtaining the proportion
of such elements in the finite alternating group
An. In both cases, we...

The material presented in this paper comprises an exposition of collaborative research which at the time of writing is still unpublished. Firstly, we discuss a statistical analysis of an algorithm for random selection of elements from groups by Adrian J. Baddeley, Charles R. Leedham-Green, Martin Firth and the first author. Next, we describe an imp...

Rahilly families of pre-difference sets have been introduced by Rahilly, Praeger, Street and Bryant as a tool for constructing symmetric designs. Using orderly generation, we construct Rahilly families for various groups up to equivalence. For each equivalence class we determine the isomorphism type of the corresponding design. Some designs may be...

In a previous paper the authors described an algorithm to determine whether a group of matrices over a finite field, generated by a given set of matrices, contains one of the classical groups or the special linear group. The algorithm was designed to work for all sufficiently large field sizes and dimensions of the matrix group. However, it did not...

In a previous paper the authors described an algorithm to determine whether a group of matrices over a finite field, generated by a given set of matrices, contains one of the classical groups or the special linear group. The algorithm was designed to work for all sufficiently large field sizes and dimensions of the matrix group. However, it did not...

We describe an algorithm for computing quotients of prime-power order for finitely presented groups and its implementation in GAP. We use the opportunity (given by the design of the GAP language) to give rather more detail about such implementations than is available outside programs. We also describe some of the impact of this implementation proce...

Representations) 1 implementing this idea. A striking application of constructive representation theory is the decomposition of matrices representing discrete signal transforms into a product of highly structured sparse matrices (realized in 1.147). This decomposition can be viewed as a fast algorithm for the signal transform. Another application i...

We describe in some detail an algorithm which is an important ingredient in the performance of programs for working with descriptions of groups of prime-power order. It is an algorithm for computing a useful power-commutator presentation for thep-covering group of a group given by a power-commutator presentation. We prove that the algorithm is corr...

The theory of vertex-transitive graphs has developed in parallel with the theory of transitive permutation groups. The chapter explores the way the two theories have influenced each other. Examples are drawn from the enumeration of vertex-transitive graphs of small order, the classification problem for finite distance transitive graphs, and the inv...

A finite soluble quotient algorithm which computes power conjugate presentations for finite soluble quotients of finitely presented groups is described. A version of this algorithm has been implemented in C and is available as the ANU Soluble Quotient Program.

An algorithm for computing power conjugate presentations for finite soluble quotients of predetermined structure of finitely presented groups is described. Practical aspects of an implementation are discussed.

The work in this thesis was carried out in the area of computational group theory. The latter is concerned with designing algorithm s and developing their practical implementations for investigating problem s regarding groups. An important class of groups are finite soluble groups. These can be described in a computationally convenient way by power...

The block-transitive point-imprimitive 2-(729,8,1) designs are classified. They all have full automorphism group of order 729.13 which is an extension of a groupN of order 729, acting regularly on points, by a group of order 13. There are, up to isomorphism, 27 designs withN elementary abelian, 13 designs withN=Z93 and 427 designs withN the relativ...

## Projects

Projects (3)

In this project we investigate properties of subgroups of odd indices in finite nonabelian simple groups.