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Cheryl E. Praeger

Publications and source records attributed to Cheryl E. Praeger.

At least 91 records · Page 5Linked to original sources
arXiv2017

A graph-theoretic description of scale-multiplicative semigroups of automorphisms

It is shown that a flat subgroup, $H$, of the totally disconnected, locally compact group $G$ decomposes into a finite number of subsemigroups on which the scale function is multiplicative. The image, $P$, of a multiplicative semigroup in the quotient, $H/H(1)$, of $H$ by its uniscalar subgroup has a unique minimal generating set which determines a natural Cayley graph structure on $P$. For each compact, open subgroup $U$ of $G$, a graph is defined and it is shown that if $P$ is multiplicative over $U$ then this graph is a regular, rooted, strongly simple $P$-graph. This extends to higher rank the result of R. Möller that $U$ is tidy for $x$ if and only if a certain graph is a regular, rooted tree.

math.GR↗
arXiv2017

Symmetric graphs with complete quotients

Let $Γ$ be a $G$-symmetric graph with vertex set $V$. We suppose that $V$ admits a $G$-partition $\mathcal{B} = \{ B_0, ... , B_b \}$, with parts of size $v$, and that the quotient graph induced on $\mathcal B$ is a complete graph of order $b+1$. Then, for each pair of distinct suffices $i, j$, the graph induced on the union $B_i\cup B_j$ is bipartite with each vertex of valency $0$ or $t$ (a constant). When $t=1$, it was shown earlier how a flag-transitive $1$-design $D(B_i)$ induced on a part $B_i$ can sometimes be used to classify possible triples $(Γ, G, \mathcal B)$. Here we extend these ideas to $t > 1$ and prove that, if the group induced by $G$ on a part $B_i$ is $2$-transitive and the "blocks" of $D(B_i)$ have size less than $v$, then either (i) $v < b$, or (ii) the triple $(Γ, G, \mathcal B)$ is known explicitly.

math.CO↗
arXiv2017

On the dimension of twisted centralizer codes

Given a field $F$, a scalar $λ\in F$ and a matrix $A\in F^{n\times n}$, the twisted centralizer code $C_F(A,λ):=\{B\in F^{n\times n}\mid AB-λBA=0\}$ is a linear code of length $n^2$. When $A$ is cyclic and $λ\ne0$ we prove that $\dim C_F(A,λ)=\mathrm{deg}(\gcd(c_A(t),λ^n c_A(λ^{-1}t)))$ where $c_A(t)$ denotes the characteristic polynomial of $A$. We also show how $C_F(A,λ)$ decomposes, and we estimate the probability that $C_F(A,λ)$ is nonzero when $|F|$ is finite. Finally, we prove $\dim C_F(A,λ)\leqslant n^2/2$ for $λ\not\in\{0,1\}$ and `almost all' matrices $A$.

math.CO↗
arXiv2017

Group factorisations, uniform automorphisms, and permutation groups of simple diagonal type

We present a new proof, which is independent of the finite simple group classification and applies also to infinite groups, that quasiprimitive permutation groups of simple diagonal type cannot be embedded into wreath products in product action. The proof uses several deep results that concern factorisations of direct products involving subdirect subgroups. We find that such factorisations are controlled by the existence of uniform automorphisms.

math.GR↗
arXiv2017

Twisted Centralizer Codes

Given an $n\times n$ matrix $A$ over a field $F$ and a scalar $a\in F$, we consider the linear codes $C(A,a):=\{B\in F^{n\times n}\mid \,AB=aBA\}$ of length $n^2$. We call $C(A,a)$ a twisted centralizer code. We investigate properties of these codes including their dimensions, minimum distances, parity-check matrices, syndromes, and automorphism groups. The minimal distance of a centralizer code (when $a=1$) is at most $n$, however for $a\ne 0,1$ the minimal distance can be much larger, as large as $n^2$.

math.CO↗
arXiv2017

Simple groups, product actions, and generalised quadrangles

The classification of flag-transitive generalised quadrangles is a long-standing open problem at the interface of finite geometry and permutation group theory. Given that all known flag-transitive generalised quadrangles are also point-primitive (up to point-line duality), it is likewise natural to seek a classification of the point-primitive examples. Working towards this aim, we are led to investigate generalised quadrangles that admit a collineation group $G$ preserving a Cartesian product decomposition of the set of points. It is shown that, under a generic assumption on $G$, the number of factors of such a Cartesian product can be at most four. This result is then used to treat various types of primitive and quasiprimitive point actions. In particular, it is shown that $G$ cannot have holomorph compound O'Nan-Scott type. Our arguments also pose purely group-theoretic questions about conjugacy classes in non-Abelian finite simple groups, and about fixities of primitive permutation groups.

math.GR↗
arXiv2017

New characterisations of the Nordstrom-Robinson codes

In his doctoral thesis, Snover proved that any binary $(m,256,δ)$ code is equivalent to the Nordstrom-Robinson code or the punctured Nordstrom-Robinson code for $(m,δ)=(16,6)$ or $(15,5)$ respectively. We prove that these codes are also characterised as \emph{completely regular} binary codes with $(m,δ)=(16,6)$ or $(15,5)$, and moreover, that they are \emph{completely transitive}. Also, it is known that completely transitive codes are necessarily completely regular, but whether the converse holds has up to now been an open question. We answer this by proving that certain completely regular codes are not completely transitive, namely, the (Punctured) Preparata codes other than the (Punctured) Nordstrom-Robinson code.

math.CO↗
arXiv2016

A normal quotient analysis for some families of oriented four-valent graphs

We analyse the normal quotient structure of several infinite families of finite connected edge-transitive, four-valent oriented graphs. These families were singled out by Marusic and others to illustrate various different internal structures for these graphs in terms of their alternating cycles (cycles in which consecutive edges have opposite orientations). Studying the normal quotients gives fresh insights into these oriented graphs: in particular we discovered some unexpected `cross-overs' between these graph families when we formed normal quotients. We determine which of these oriented graphs are `basic', in the sense that their only proper normal quotients are degenerate. Moreover, we show that the three types of edge-orientations studied are the only orientations, of the underlying undirected graphs in these families, which are invariant under a group action which is both vertex-transitive and edge-transitive.

math.CO↗
arXiv2016

Finite edge-transitive oriented graphs of valency four with cyclic normal quotients

We study finite four-valent graphs Gamma admitting an edge-transitive group G of automorphisms such that G determines and preserves an edge-orientation on Gamma, and such that at least one G-normal quotient is a cycle (a quotient modulo the orbits of a normal subgroup of G). We show on the one hand that the number of distinct cyclic G-normal quotients can be unboundedly large. On the other hand existence of independent cyclic G-normal quotients (that is, they are not extendable to a common cyclic G-normal quotient) places severe restrictions on the graph Gamma and we classify all examples. We show there are five infinite families of such pairs (Gamma, G), and in particular that all such graphs have at least one normal quotient which is an unoriented cycle. We compare this new approach with existing treatments for the sub-class of weak metacirculant graphs with these properties, finding that only two infinite families of examples occur in common from both analyses. Several open problems are posed.

math.CO↗
arXiv2016

Primitive prime divisors and the $n$-th cyclotomic polynomial

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 Hering. Given positive constants $c$ and $k$, we give an algorithm for determining all pairs $(n,q)$ with $Φ^*_n(q)\le cn^k$. This algorithm is used to extend (and correct) a result of Hering which is useful for classifying certain families of subgroups of finite linear groups.

math.NT↗
arXiv2016

On the Complexity of Multiplication in the Iwahori--Hecke Algebra of the Symmetric Group

We present new efficient data structures for elements of Coxeter groups of type $A_m$ and their associated Iwahori--Hecke algebras $H(A_m)$. Usually, elements of $H(A_m)$ are represented as simple coefficient list of length $M = (m+1)!$ with respect to the standard basis, indexed by the elements of the Coxeter group. In the new data structure, elements of $H(A_m)$ are represented as nested coefficient lists. While the cost of addition is the same in both data structures, the new data structure leads to a huge improvement in the cost of multiplication in~$H(A_m)$.

math.GR↗
arXiv2016

Point-primitive, line-transitive generalised quadrangles of holomorph type

Let $G$ be a group of collineations of a finite thick generalised quadrangle $Γ$. Suppose that $G$ acts primitively on the point set $\mathcal{P}$ of $Γ$, and transitively on the lines of $Γ$. We show that the primitive action of $G$ on $\mathcal{P}$ cannot be of holomorph simple or holomorph compound type. In joint work with Glasby, we have previously classified the examples $Γ$ for which the action of $G$ on $\mathcal{P}$ is of affine type. The problem of classifying generalised quadrangles with a point-primitive, line-transitive collineation group is therefore reduced to the case where there is a unique minimal normal subgroup $M$ and $M$ is non-Abelian.

math.GR↗
arXiv2016

Conway's groupoid and its relatives

In 1997, John Conway constructed a $6$-fold transitive subset $M_{13}$ of permutations on a set of size $13$ for which the subset fixing any given point was isomorphic to the Mathieu group $M_{12}$. The construction was via a "moving-counter puzzle" on the projective plane ${\rm PG}(2,3)$. We discuss consequences and generalisations of Conway's construction. In particular we explore how various designs and hypergraphs can be used instead of ${\rm PG}(2,3)$ to obtain interesting analogues of $M_{13}$; we refer to these analogues as Conway groupoids. A number of open questions are presented.

math.GR↗
arXiv2016

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

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 to $x$ a doubly transitive action with socle the projective group $L_{n_x}(q_x)$ for an integer $n_x$ and a prime power $q_x$. It is $G$-locally projective if in addition $G$ is vertex transitive. A theorem of Trofimov reduces the classification of the $G$-locally projective graphs to the case where the distance factors are as in one of the known examples. Although an analogue of Trofimov's result is not yet available for weakly locally projective graphs, we would like to begin a program of characterising some of the remarkable examples. We show that if a graph is weakly locally projective with each $q_x =2$ and $n_x = 2$ or $3$, and if the distance factors are as in the examples arising from the rank 3 tilde geometries of the groups $M_{24}$ and $He$, then up to isomorphism there are exactly two possible amalgams. Moreover, we consider an infinite family of amalgams of type $\mathcal{U}_n$ (where each $q_x=2$ and $n=n_x+1\geq 4$) and prove that if $n\geq 5$ there is a unique amalgam of type $\mathcal{U}_n$ and it is unfaithful, whereas if $n=4$ then there are exactly four amalgams of type $\mathcal{U}_4$, precisely two of which are faithful, namely the ones related to $M_{24}$ and $He$, and one other which has faithful completion $A_{16}$.

math.GR↗
arXiv2016

Base sizes of imprimitive linear groups and orbits of general linear groups on spanning tuples

For a subgroup $L$ of the symmetric group $S_\ell$, we determine the minimal base size of $GL_d(q)\wr L$ acting on $V_d(q)^\ell$ as an imprimitive linear group. This is achieved by computing the number of orbits of $GL_d(q)$ on spanning $m$-tuples, which turns out to be the number of $d$-dimensional subspaces of $V_m(q)$. We then use these results to prove that for certain families of subgroups $L$, the affine groups whose stabilisers are large subgroups of $GL_d(q)\wr L$ satisfy a conjecture of Pyber concerning bases.

math.GR↗
arXiv2016

Affine primitive symmetric graphs of diameter two

Let $n$ be a positive integer, $q$ be a prime power, and $V$ be a vector space of dimension $n$ over $\mathbb{F}_q$. Let $G := V \rtimes G_0$, where $G_0$ is an irreducible subgroup of ${\rm GL}(V)$ which is maximal by inclusion with respect to being intransitive on the set of nonzero vectors. We are interested in the class of all diameter two graphs $Γ$ that admit such a group $G$ as an arc-transitive, vertex-quasiprimitive subgroup of automorphisms. In particular, we consider those graphs for which $G_0$ is a subgroup of either ${\rm ΓL}(n,q)$ or ${\rm ΓSp}(n,q)$ and is maximal in one of the Aschbacher classes $\mathcal{C}_i$, where $i \in \{2,4,5,6,7,8\}$. We are able to determine all graphs $Γ$ which arise from $G_0 \leq {\rm ΓL}(n,q)$ with $i \in \{2,4,8\}$, and from $G_0 \leq {\rm ΓSp}(n,q)$ with $i \in \{2,8\}$. For the remaining classes we give necessary conditions in order for $Γ$ to have diameter two, and in some special subcases determine all $G$-symmetric diameter two graphs.

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