Searcharxiv⌕ Search

arXiv subjects

Michael Giudici

Publications and source records attributed to Michael Giudici.

At least 55 records · Page 3Linked to original sources
arXiv2017

Locally elusive classical groups

Let $G$ be a transitive permutation group of degree $n$ with point stabiliser $H$ and let $r$ be a prime divisor of $n$. We say that $G$ is $r$-elusive if it does not contain a derangement of order $r$. The problem of determining the $r$-elusive primitive groups can be reduced to the almost simple case, and the purpose of this paper is to complete the study of $r$-elusivity for almost simple classical groups. Building on our earlier work for geometric actions of classical groups, in this paper we handle the remaining non-geometric actions where $H$ is almost simple and irreducible. This requires a completely different approach, using tools from the representation theory of quasisimple groups.

math.GR↗
arXiv2016

Arc-transitive digraphs with quasiprimitive local actions

Let $Γ$ be a finite $G$-vertex-transitive digraph. The in-local action of $(Γ,G)$ is the permutation group $L_-$ induced by the vertex-stabiliser on the set of in-neighbours of $v$. The out-local action $L_+$ is defined analogously. Note that $L_-$ and $L_+$ may not be isomorphic. We thus consider the problem of determining which pairs $(L_-,L_+)$ are possible. We prove some general results, but pay special attention to the case when $L_-$ and $L_+$ are both quasiprimitive. (Recall that a permutation group is quasiprimitive if each of its nontrivial normal subgroups is transitive.) Along the way, we prove a structural result about pairs of finite quasiprimitive groups of the same degree, one being (abstractly) isomorphic to a proper quotient of the other.

math.GR↗
arXiv2016

A theory of semiprimitive groups

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 arising in universal algebra related to collapsing monoids and the graph-restrictive problem for permutation groups. Here we develop a theory of semiprimitive groups which encompasses their structure, their quotient actions and a method by which all finite semiprimitive groups are constructed. We also extend some results from the theory of primitive groups to semiprimitive groups, and conclude with open problems of a similar nature.

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

Cubic arc-transitive $k$-circulants

For an integer $k\geq 1$, a graph is called a $k$-circulant if its automorphism group contains a cyclic semiregular subgroup with $k$ orbits on the vertices. We show that, if $k$ is even, there exist infinitely many cubic arc-transitive $k$-circulants. We conjecture that, if $k$ is odd, then a cubic arc-transitive $k$-circulant has order at most $6k^2$. Our main result is a proof of this conjecture when $k$ is squarefree and coprime to $6$.

math.CO↗
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.

math.CO↗
arXiv2015

Realizability problem for commuting graphs

We investigate properties which ensure that a given finite graph is the commuting graph of a group or semigroup. We show that all graphs on at least two vertices such that no vertex is adjacent to all other vertices is the commuting graph of some semigroup. Moreover, we obtain a complete classification of the graphs with an isolated vertex or edge that are the commuting graph of a group and the cycles that are the commuting graph of a centrefree semigroup.

math.GR↗
arXiv2015

On locally semiprimitive graphs and a theorem of Weiss

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 semiprimitive group is a generalisation of an affine group. We show that if the graph has valency coprime to six, then there is a bound on the order of the vertex stabilisers depending on the valency alone. We also prove a detailed structure theorem for the vertex stabilisers in the remaining case. This is a contribution to an ongoing project to investigate the validity of the Potočnik-Spiga-Verret Conjecture.

math.GR↗
arXiv2014

Entry-Faithful $2$-Neighbour Transitive Codes

We consider a code to be a subset of the vertex set of a Hamming graph. The set of $s$-neighbours of a code is the set of vertices, not in the code, at distance $s$ from some codeword, but not distance less than $s$ from any codeword. A $2$-neighbour transitive code is a code which admits a group $X$ of automorphisms which is transitive on the $s$-neighbours, for $s=1,2$, and transitive on the code itself. We give a classification of $2$-neighbour transitive codes, with minimum distance $δ\geq 5$, for which $X$ acts faithfully on the set of entries of the Hamming graph.

math.CO↗
arXiv2014

Elusive groups of automorphisms of digraphs of small valency

A transitive permutation group is called elusive if it contains no semiregular element. We show that no group of automorphisms of a connected graph of valency at most four is elusive and determine all the elusive groups of automorphisms of connected digraphs of out-valency at most three.

math.CO↗
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

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

A class of semiprimitive groups that are graph-restrictive

We prove that an infinite family of semiprimitive groups are graph-restrictive. This adds to the evidence for the validity of the PSV Conjecture and increases the minimal imprimitive degree for which this conjecture is open to 12. Our result can be seen as a generalisation of the well-known theorem of Tutte on cubic graphs. The proof uses the amalgam method, adapted to this new situation.

math.GR↗
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

Locally s-distance transitive graphs and pairwise transitive designs

The study of locally s-distance transitive graphs initiated by the authors in previous work, identified that graphs with a star quotient are of particular interest. This paper shows that the study of locally s-distance transitive graphs with a star quotient is equivalent to the study of a particular family of designs with strong symmetry properties that we call nicely affine and pairwise transitive. We show that a group acting regularly on the points of such a design must be abelian and give a general construction for this case.

math.CO↗
arXiv2013

Semiregular automorphisms of edge-transitive graphs

The polycirculant conjecture asserts that every vertex-transitive digraph has a semiregular automorphism, that is, a nontrivial automorphism whose cycles all have the same length. In this paper we investigate the existence of semiregular automorphisms of edge-transitive graphs. In particular, we show that any regular edge-transitive graph of valency three or four has a semiregular automorphism.

math.CO↗
arXiv2013

Characterising star-transitive and st(edge)-transitive graphs

Recent work of Lazarovich provides necessary and sufficient conditions on a graph L for there to exist a unique simply-connected (k,L)-complex. The two conditions are symmetry properties of the graph, namely star-transitivity and st(edge)-transitivity. In this paper we investigate star-transitive and st(edge)-transitive graphs by studying the structure of the vertex and edge stabilisers of such graphs. We also provide new examples of graphs that are both star-transitive and st(edge)-transitive.

math.CO↗