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Michael Giudici

Publications and source records attributed to Michael Giudici.

At least 19 recordsLinked to original sources
arXiv2025

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.

math.GR↗
arXiv2025

Finite permutation groups with quasi-semiregular elements

A quasi-semiregular element in a permutation group is an element that has a unique fixed point and acts semiregularly on the remaining points. Such elements were first studied in the context of automorphisms of graphs and occur naturally in many families of permutation groups, such as Frobenius and Zassenhaus groups. They also arise in the context of groups with a strongly $p$-embedded subgroup. We investigate the question of which finite permutation groups contain quasi-semiregular elements, with particular attention to the primitive permutation groups. We determine the O'Nan-Scott types of primitive groups that can contain quasi-semiregular elements and reduce the question to the affine and almost simple cases. In the almost simple case, we obtain a complete classification when the socle is alternating or sporadic.

math.GR↗
arXiv2024

Prime power coverings of groups

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 $|G:U|$ is bounded by some function of $|A:G|$, and this conjecture has number theoretic implications for relative Brauer groups of algebraic number fields. 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 case where $U$ is a maximal subgroup of $G$. The proof uses a new bound on the order of a nonabelian finite simple group in terms of its number of classes of elements of prime power order, which in turn depends on the Classification of the Finite Simple Groups.

math.GR↗
arXiv2024

Finite simple groups have many classes of $p$-elements

For an element $x$ of a finite group $T$, the $\mathrm{Aut}(T)$-class of $x$ is the set $\{ x^\sigma\mid \sigma\in \mathrm{Aut}(T)\}$. 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 $\mathrm{Aut}(T)$-classes of elements of $T$ of $p$-power order. This bound is a substantial generalisation of results of Pyber, and of H\'ethelyi and K\"ulshammer, and it has implications for relative Brauer groups of finite extensions of global fields.

math.GR↗
arXiv2024

Vertex-primitive s-arc-transitive digraphs of symplectic groups

A digraph is $s$-arc-transitive if its automorphism group is transitive on directed paths with $s$ edges, that is, on $s$-arcs. Although infinite families of finite $s$-arc transitive digraphs of arbitrary valency were constructed by the third author in 1989, existence of a vertex-primitive $2$-arc-transitive digraph was not known until an infinite family was constructed by the second author with Li and Xia in 2017. This led to a conjecture by the second author and Xia in 2018 that, for a finite vertex-primitive $s$-arc-transitive digraph, $s$ is at most $2$, together with their proof that it is sufficient to prove the conjecture for digraphs with an almost simple group of automorphisms. This paper confirms the conjecture for finite symplectic groups.

math.CO↗
arXiv2024

Tactical decompositions in finite polar spaces and non-spreading classical group actions

For finite classical groups acting naturally on the set of points of their ambient polar spaces, the symmetry properties of \emph{synchronising} and \emph{separating} are equivalent to natural and well-studied problems on the existence of certain configurations in finite geometry. The more general class of \emph{spreading} permutation groups is harder to describe, and it is the purpose of this paper to explore this property for finite classical groups. In particular, we show that for most finite classical groups, their natural action on the points of its polar space is non-spreading. We develop and use a result on tactical decompositions (an \emph{AB-Lemma}) that provides a useful technique for finding witnesses for non-spreading permutation groups. We also consider some of the other primitive actions of the classical groups.

math.GR↗
arXiv2024

Bounding $s$ for vertex-primitive $s$-arc-transitive digraphs of alternating and symmetric groups

Determining an upper bound on $s$ for finite vertex-primitive $s$-arc-transitive digraphs has received considerable attention dating back to a question of Praeger in 1990. It was shown by Giudici and Xia that the smallest upper bound on $s$ is attained for some digraph admitting an almost simple $s$-arc-transitive group. In this paper, based on the work of Pan, Wu and Yin, we prove that $s\leqslant 2$ in the case where the group is an alternating or symmetric group.

math.CO↗
arXiv2023

New 2-closed groups that are not automorphism groups of digraphs

In this paper we extend the construction of Giudici, Morgan and Zhou [arXiv:2110.07896] to give the first known examples of nonregular, $2$-closed permutation groups of rank greater than $4$ that are not the automorphism group of any digraph. We also show that this construction only gives examples for four particular primes.

math.GR↗
arXiv2023

Spreading primitive groups of diagonal type do not exist

The synchronisation hierarchy of finite permutation groups consists of classes of groups lying between 2-transitive groups and primitive groups. This includes the class of spreading groups, which are defined in terms of sets and multisets of permuted points, and which are known to be primitive of almost simple, affine or diagonal type. In this paper, we prove that in fact no spreading group of diagonal type exists. As part of our proof, we show that all non-abelian finite simple groups, other than six sporadic groups, have a transitive action in which a proper normal subgroup of a point stabiliser is supplemented by all corresponding two-point stabilisers.

math.GR↗
arXiv2023

Separating rank 3 graphs

We classify, up to some notoriously hard cases, the rank 3 graphs which fail to meet either the Delsarte or the Hoffman bound. As a consequence, we resolve the question of separation for the corresponding rank 3 primitive groups and give new examples of synchronising, but not $\mathbb{Q}\mathrm{I}$, groups of affine type.

math.CO↗
arXiv2023

Vertex-primitive s-arc-transitive digraphs admitting a Suzuki or Ree group

The investigation of s-arc-transitivity of digraphs can be dated back to 1989 when the third author showed that s can be arbitrarily large if the action on vertices is imprimitive. However, the situation is completely different when the digraph is vertex-primitive and not a directed cycle. In 2017 the second author, Li and Xia constructed the first infinite family of G-vertex-primitive 2-arc-transitive examples, and asked if there is an upper bound on s for G-vertex-primitive s-arc-transitive digraphs w=that are not directed. In 2018 the second author and Xia showed that if there is a largest such value of s then it will occur when G is almost simple. So far it has been shown that s\leq 2 for almost simple groups whose socle is an alternating group or a projective special linear group. The contribution of this paper is to prove that s\leq 1 in the case of the Suzuki and the small Ree groups. We give constructions with s=1 to show that the bound is sharp.

math.GR↗
arXiv2023

Total closure for permutation actions of finite nonabelian simple groups

For a positive integer $k$, a group $G$ is said to be totally $k$-closed if for each set $Ω$ upon which $G$ acts faithfully, $G$ is the largest subgroup of $\mathrm{Sym}(Ω)$ that leaves invariant each of the $G$-orbits in the induced action on $Ω\times\cdots\times Ω=Ω^k$. Each finite group $G$ is totally $|G|$-closed, and $k(G)$ denotes the least integer $k$ such that $G$ is totally $k$-closed. We address the question of determining the closure number $k(G)$ for finite simple groups $G$. Prior to our work it was known that $k(G)=2$ for cyclic groups of prime order and for precisely six of the sporadic simple groups, and that $k(G)\geq3$ for all other finite simple groups. We determine the value for the alternating groups, namely $k(A_n)=n-1$. In addition, for all simple groups $G$, other than alternating groups and classical groups, we show that $k(G)\leq 7$. Finally, if $G$ is a finite simple classical group with natural module of dimension $n$, we show that $k(G)\leq n+2$ if $n \ge 14$, and $k(G) \le \lfloor n/3 + 12 \rfloor$ otherwise, with smaller bounds achieved by certain families of groups. This is achieved by determining a uniform upper bound (depending on $n$ and the type of $G$) on the base sizes of the primitive actions of $G$, based on known bounds for specific actions. We pose several open problems aimed at completing the determination of the closure numbers for finite simple groups.

math.GR↗
arXiv2022

On primitive $2$-closed permutation groups of rank at most four

We characterise the primitive 2-closed groups $G$ of rank at most four that are not the automorphism group of a graph or digraph and show that if the degree is at least 2402 then there are just two infinite families or $G\leqslant \mathrm{A}Γ\mathrm{L}_1(p^d)$, the 1-dimensional affine semilinear group. These are the first known examples of non-regular 2-closed groups that are not the automorphism group of a graph or digraph.

math.CO↗
arXiv2022

A generalization of Szep's conjecture for almost simple groups

We prove a natural generalization of Szep's conjecture. Given an almost simple group $G$ with socle not isomorphic to an orthogonal group having Witt defect zero, we classify all possible group elements $x,y\in G\setminus\{1\}$ with $G={\bf N}_G (\langle x\rangle){\bf N}_G(\langle y\rangle)$, where we are denoting by ${\bf N}_G(\langle x\rangle)$ and by ${\bf N}_G(\langle y\rangle)$ the normalizers of the cyclic subgroups $\langle x\rangle$ and $\langle y\rangle$. As a consequence of this result, we classify all possible group elements $x,y\in G\setminus\{1\}$ with $G={\bf C}_G(x){\bf C}_G(y)$.

math.GR↗
arXiv2022

Synchronising primitive groups of diagonal type exist

Every synchronising permutation group is primitive and of one of three types: affine, almost simple, or diagonal. We exhibit the first known example of a synchronising diagonal type group. More precisely, we show that $\mathrm{PSL}(2,q)\times \mathrm{PSL}(2,q)$ acting in its diagonal action on $\mathrm{PSL}(2,q)$ is separating, and hence synchronising, for $q=13$ and $q=17$. Furthermore, we show that such groups are non-spreading for all prime powers $q$.

math.GR↗
arXiv2022

Locally s-arc-transitive graphs arising from product action

We study locally $s$-arc-transitive graphs arising from the quasiprimitive product action (PA). We prove that, for any locally $(G,2)$-arc-transitive graph with $G$ acting quasiprimitively with type PA on both $G$-orbits of vertices, the group $G$ does not act primitively on either orbit. Moreover, we construct the first examples of locally $s$-arc-transitive graphs of PA type that are not standard double covers of $s$-arc-transitive graphs of PA type, answering the existence question for these graphs.

math.CO↗
arXiv2022

Arc-transitive bicirculants

In this paper, we characterise the family of finite arc-transitive bicirculants. We show that every finite arc-transitive bicirculant is a normal $r$-cover of an arc-transitive graph that lies in one of eight infinite families or is one of seven sporadic arc-transitive graphs. Moreover, each of these "basic" graphs is either an arc-transitive bicirculant or an arc-transitive circulant, and each graph in the latter case has an arc-transitive bicirculant normal $r$-cover for some integer $r$.

math.CO↗