arXiv subjects

Cheryl E. Praeger

Publications and source records attributed to Cheryl E. Praeger.

At least 127 records · Page 7Linked to original sources

arXiv2014

Proportion of cyclic matrices in maximal reducible matrix algebras

Let ${\rm M}(V)={\rm M}(n,\mathbb{F}_q)$ denote the algebra of $n\times n$ matrices over $\mathbb{F}_q$, and let ${\rm M}(V)_U$ denote the (maximal reducible) subalgebra that normalizes a given $r$-dimensional subspace $U$ of $V=\mathbb{F}_q^n$ where $0<r<n$. We prove that the density of non-cyclic matrices in ${\rm M}(V)_U$ is at least $q^{-2}\left(1+c_1q^{-1}\right)$, and at most $q^{-2}\left(1+c_2q^{-1}\right)$, where $c_1$ and $c_2$ are constants independent of $n,r$, and $q$. The constants $c_1=-\frac43$ and $c_2=\frac{35}3$ suffice.

math.RA↗

arXiv2014

Triple factorisations of the general linear group and their associated geometries

Triple factorisations of finite groups $G$ of the form $G=PQP$ are essential in the study of Lie theory as well as in geometry. Geometrically, each triple factorisation $G=PQP$ corresponds to a $G$-flag transitive point/line geometry such that `each pair of points is incident with at least one line'. We call such a geometry \emph{collinearly complete}, and duality (interchanging the roles of points and lines) gives rise to the notion of \emph{concurrently complete} geometries. In this paper, we study triple factorisations of the general linear group $\mathrm{GL}(V)$ as $PQP$ where the subgroups $P$ and $Q$ either fix a subspace or fix a decomposition of $V$ as $V_1\oplus V_2$ with $\dim(V_{1})=\dim(V_{2})$.

math.GR↗

arXiv2014

Characterisation of a family of neighbour transitive codes

We consider codes of length $m$ over an alphabet of size $q$ as subsets of the vertex set of the Hamming graph $Γ=H(m,q)$. A code for which there exists an automorphism group $X\leq Aut(Γ)$ that acts transitively on the code and on its set of neighbours is said to be neighbour transitive, and were introduced by the authors as a group theoretic analogue to the assumption that single errors are equally likely over a noisy channel. Examples of neighbour transitive codes include the Hamming codes, various Golay codes, certain Hadamard codes, the Nordstrom Robinson codes, certain permutation codes and frequency permutation arrays, which have connections with powerline communication, and also completely transitive codes, a subfamily of completely regular codes, which themselves have attracted a lot of interest. It is known that for any neighbour transitive code with minimum distance at least 3 there exists a subgroup of $X$ that has a $2$-transitive action on the alphabet over which the code is defined. Therefore, by Burnside's theorem, this action is of almost simple or affine type. If the action is of almost simple type, we say the code is alphabet almost simple neighbour transitive. In this paper we characterise a family of neighbour transitive codes, in particular, the alphabet almost simple neighbour transitive codes with minimum distance at least $3$, and for which the group $X$ has a non-trivial intersection with the base group of $Aut(Γ)$. If $C$ is such a code, we show that, up to equivalence, there exists a subcode $Δ$ that can be completely described, and that either $C=Δ$, or $Δ$ is a neighbour transitive frequency permutation array and $C$ is the disjoint union of $X$-translates of $Δ$. We also prove that any finite group can be identified in a natural way with a neighbour transitive code.

math.CO↗

arXiv2014

Decomposing modular tensor products: `Jordan partitions', their parts and p-parts

Determining the Jordan canonical form of the tensor product of Jordan blocks has many applications including to the representation theory of algebraic groups, and to tilting modules. Although there are several algorithms for computing this decomposition in literature, it is difficult to predict the output of these algorithms. We call a decomposition of the form $J_r\otimes J_s=J_{λ_1}\oplus\cdots\oplus J_{λ_b}$ a `Jordan partition'. We prove several deep results concerning the $p$-parts of the $λ_i$ where $p$ is the characteristic of the underlying field. Our main results include the proof of two conjectures made by McFall in 1980, and the proof that ${\rm lcm}(r,s)$ and $\gcd(λ_1,\dots,λ_b)$ have equal $p$-parts. Finally, we establish some explicit formulas for Jordan partitions when $p=2$.

math.RT↗

arXiv2014

Elements in finite classical groups whose powers have large 1-Eigenspaces

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 characteristic.

math.GR↗

arXiv2014

Completely Transitive Designs

We view a design $\mathcal{D}$ as a set of $k$-subsets of a fixed set $X$ of $v$ points. A $k$-subset of $X$ is at distance $i$ from $\mathcal{D}$ if it intersects some $k$-set in $\mathcal{D}$ in $k-i$ points, and no subset in more than $k-i$ points. Thus $\mathcal{D}$ determines a partition by distance of the $k$-subsets of $X$. We say $\mathcal{D}$ is completely transitive if the cells of this partition are the orbits of the automorphism group of $\mathcal{D}$ in its induced action on the $k$-subsets of $X$. This paper initiates a study of completely transitive designs $\mathcal{D}$. A classification is given of all examples for which the automorphism group is not primitive on $X$. In the primitive case the focus is on examples with the property that any two distinct $k$-subsets in $\mathcal{D}$ have at most $k-3$ points in common. Here a reduction is given to the case where the automorphism group is $2$-transitive on $X$. New constructions are given by classifying all examples for some families of $2$-transitive groups, leaving several unresolved cases.

math.CO↗

arXiv2014

Elusive Codes in Hamming Graphs

We consider a code to be a subset of the vertex set of a Hamming graph. We examine elusive pairs, code-group pairs where the code is not determined by knowledge of its set of neighbours. We construct a new infinite family of elusive pairs, where the group in question acts transitively on the set of neighbours of the code. In our examples, we find that the alphabet size always divides the length of the code, and prove that there is no elusive pair for the smallest set of parameters for which this is not the case. We also pose several questions regarding elusive pairs.

math.CO↗

arXiv2014

Generation of finite classical groups by pairs of elements with large fixed point spaces

We study `good elements' in finite $2n$-dimensional classical groups $G$: namely $t$ is a `good element' if $o(t)$ is divisible by a primitive prime divisor of $q^n-1$ for the relevant field order $q$, and $t$ fixes pointwise an $n$-space. The group ${\rm{SL}}_{2n}(q)$ contains such elements, and they are present in ${\rm{Su}}_{2n}(q), {\rm{Sp}}_{2n}(q), {\rm{So}}^ε_{2n}(q)$, only if $n$ is odd, even, even, respectively. We prove that there is an absolute positive constant $c$ such that two random conjugates of $t$ generate $G$ with probability at least $c$, if $G\ne {\rm{Sp}}_{2n}(q)$ with $q$ even. In the exceptional case $G={\rm{Sp}}_{2n}(q)$ with $q$ even, two conjugates of $t$ never generate $G$: in this case we prove that two random conjugates of $t$ generate a subgroup ${\rm{SO}}^ε_{2n}(q)$ with probability at least $c$. The results (proved for all field orders at least $4$) underpin analysis of new constructive recognition algorithms for classical groups in even characteristic, which succeed where methods utilising involution centralisers are not available.

math.GR↗

arXiv2014

Nilpotent-independent sets and estimation in matrix algebras

Efficient methods for computing with matrices over finite fields often involve randomised algorithms, where matrices with a certain property are sought via repeated random selection. Complexity analyses for these algorithms require knowledge of the proportion of relevant matrices in the ambient group or algebra. We introduce a method for estimating proportions of families $N$ of elements in the algebra of all $d\times d$ matrices over a field of order $q$, where membership of a matrix in $N$ depends only on its `invertible part'. The method is based on estimating proportions of certain subsets of $\text{GL}(d,q)$ depending on $N$, so that existing estimation techniques for nonsingular matrices can be leveraged to deal with families containing singular matrices. As an application we investigate primary cyclic matrices, which are used in the Holt-Rees MEAT-AXE algorithm for testing irreducibility of matrix algebras.

math.GR↗

arXiv2014

Diagonally Neighbour Transitive Codes and Frequency Permutation Arrays

Constant composition codes have been proposed as suitable coding schemes to solve the narrow band and impulse noise problems associated with powerline communication. In particular, a certain class of constant composition codes called frequency permutation arrays have been suggested as ideal, in some sense, for these purposes. In this paper we characterise a family of neighbour transitive codes in Hamming graphs in which frequency permutation arrays play a central rode. We also classify all the permutation codes generated by groups in this family.

math.CO↗

arXiv2014

Twisted Permutation Codes

We introduce twisted permutation codes, which are frequency permutation arrays analogous to repetition permutation codes, namely, codes obtained from the repetition construction applied to a permutation code. In particular, we show that a lower bound for the minimum distance of a twisted permutation code is the minimum distance of a repetition permutation code. We give examples where this bound is tight, but more importantly, we give examples of twisted permutation codes with minimum distance strictly greater than this lower bound.

math.CO↗

arXiv2014

Arithmetic results on orbits of linear groups

Let $p$ be a prime and $G$ a subgroup of $GL_d(p)$. We define $G$ to be $p$-exceptional if it has order divisible by $p$, but all its orbits on vectors have size coprime to $p$. We obtain a classification of $p$-exceptional linear groups. This has consequences for a well known conjecture in representation theory, and also for a longstanding question concerning 1/2-transitive linear groups (i.e. those having all orbits on nonzero vectors of equal length), classifying those of order divisible by $p$.

math.GR↗

arXiv2014

Two-sided Cayley graphs

We introduce a family of graphs that generalises the class of Cayley graphs. For non-empty subsets L, R of a group G, the two-sided Cayley graph 2SC(G;L,R) is the directed graph with vertex set G and an arc from x to y if and only if y=a^{-1}xb for some a in L and b in R. Thus, in common with Cayley graphs, two-sided Cayley graphs may be useful to model networks as the same routing and communication scheme can be implemented at each vertex. We determine when two-sided Cayley graphs are simple undirected graphs, and give sufficient conditions for them to be connected, vertex-transitive, or Cayley graphs. Several open problems are posed. Many examples are given, including one on 12 vertices with connected components of sizes 4 and 8.

math.CO↗

arXiv2014

Normal Edge-Transitive Cayley Graphs of Frobenius Groups

A Cayley Graph for a group $G$ is called normal edge-transitive if it admits an edge-transitive action of some subgroup of the Holomorph of $G$ (the normaliser of a regular copy of $G$ in $\operatorname{Sym}(G)$). We complete the classification of normal edge-transitive Cayley graphs of order a product of two primes by dealing with Cayley graphs for Frobenius groups of such orders. We determine the automorphism groups of these graphs, proving in particular that there is a unique vertex-primitive example, namely the flag graph of the Fano plane.

math.CO↗

arXiv2014

Primary Cyclic Matrices in Irreducible Matrix Subalgebras

Primary Cyclic matrices were used (but not named) by Holt and Rees in their version of Parker's MEAT-AXE algorithm to test irreducibility of finite matrix groups and algebras. They are matrices $X$ with at least one cyclic component in the primary decomposition of the underlying vector space as an $X$-module. Let $\operatorname{M}(c,q^b)$ be an irreducible subalgebra of $\operatorname{M}(n,q)$, where $n=bc >c$. We prove a generalisation of the Kung-Stong Cycle Index, and use it to obtain a lower bound for the proportion of primary cyclic matrices in $\operatorname{M}(c,q^b)$. This extends work of Glasby and the second author on the case $b=1$.

math.CO↗

arXiv2013

Finite primitive permutation groups and regular cycles of their elements

We conjecture that if $G$ is a finite primitive group and if $g$ is an element of $G$, then either the element $g$ has a cycle of length equal to its order, or for some $r,m$ and $k$, the group $G\leq S_m\wr S_r$, preserving a product structure of $r$ direct copies of the natural action of $S_m$ or $A_m$ on $k$-sets. In this paper we reduce this conjecture to the case that $G$ is an almost simple group with socle a classical group.

math.GR↗

arXiv2013

Neighbour-transitive codes in Johnson graphs

The Johnson graph J(v,k) has, as vertices, the k-subsets of a v-set V, and as edges the pairs of k-subsets with intersection of size k-1. We introduce the notion of a neighbour-transitive code in J(v,k). This is a vertex subset Γsuch that the subgroup G of graph automorphisms leaving Γinvariant is transitive on both the set Γof `codewords' and also the set of `neighbours' of Γ, which are the non-codewords joined by an edge to some codeword. We classify all examples where the group G is a subgroup of the symmetric group on V and is intransitive or imprimitive on the underlying v-set V. In the remaining case where G lies in Sym(V) and G is primitive on V, we prove that, provided distinct codewords are at distance at least 3 in J(v,k), then G is 2-transitive on V. We examine many of the infinite families of finite 2-transitive permutation groups and construct surprisingly rich families of examples of neighbour-transitive codes. A major unresolved case remains.

math.CO↗

arXiv2013

Conjectures on the normal covering number of the finite symmetric and alternating groups

Let $γ(S_n)$ be the minimum number of proper subgroups $H_i$ of the symmetric group $S_n$ such that each element in $S_n$ lies in some conjugate of one of the $H_i.$ In this paper we conjecture that $$γ(S_n)=\frac{n}{2}\left(1-\frac{1}{p_1}\right) \left(1-\frac{1}{p_2}\right)+2,$$ where $p_1,p_2$ are the two smallest primes in the factorization of $n$ and $n$ is neither a prime power nor a product of two primes. Support for the conjecture is given by a previous result for $n=p_1^{α_1}p_2^{α_2},$ with $(α_1,α_2)\neq (1,1)$. We give further evidence by confirming the conjecture for integers of the form $n=15q$ for an infinite set of primes $q$, and by reporting on a Magma computation. We make a similar conjecture for $γ(A_n)$, when $n$ is even, and provide a similar amount of evidence.

math.GR↗